(The much brief answer by @annav is good; readers should please look at it first.)
This long answer is only for folks interesting in understanding in detail why isospin has such an odd and non-intuitive relationship to electric charge, and how that led to the Gell-Mann - Nishijima formula
One of the more unexpected findings of particle physics over the last century is that the magnitude of the electric charge, which is labeled $1 e$ and is the same as the charge of a proton, links together various pairs of fundamental and composite fermions in peculiar and unexpected ways, one of them being similar particle masses. These pairs that differ by ${\pm}1 e$ appear to be connected some deeper fashion that causes them to share other important properties, including the ability to transform into each other via the weak interaction.
The history of this observation is unfortunately both decades in the making and historically quite messy, since the observation itself grew and evolved as more particles were found. The biggest changes came when the Standard Model of particle physics evolved, and with it the concept of fractionally charged fermions called quarks. Quarks are important to this charge separation rule not just because many other particles (including neutrons and protons) are composed of quarks, but in particular because the quarks also follow the rule that particles separated by one unit of electric charge may be more deeply linked together. That really is a bit surprising if you think about it, since quarks don't even has single-unit charges! The very common up quark $u$ for example has a charge of $+\frac{2}{3} e$, and the similarly very common down quark $d$ has a change of $-\frac{1}{3} e$.
But notice something about those two fractional charge values: Just how large is the separation between $u$ and $d$ in terms of electric charge? Well, it's just $(+\frac{2}{3}) - (-\frac{1}{3}) = 1 e$. That is, the $u$ and $d$ quarks are separated by exactly one of unit electric charge. So, since I started this discussion by saying that fermions separated by one unit of electric charge are sometimes oddly linked, is this true for $u$ and $d$? Do they in fact share any important and unexpected properties, such as similar masses?
The answer is yes: The $u$ and $d$ quarks are clearly linked in a rather profound way by this separation of one charge unit, to the extent that they almost seem to be the same particle in two different states of electric charge, either $+\frac{2}{3} e$ or $-\frac{1}{3} e$. (For completeness, the other most important example of the one-charge separation pair is the electron and its neutrino. That is for some other answer.)
Why would I say that $u$ and $d$ are almost like two states of the same particle?
Well, for one thing, their masses as expressed within particles such as protons and neutrons are strikingly similar. Physicists have found over the last century that such coincidences of mass should never be ignored, even if they do eventually prove to be accidental. In this particular case it is no accident. The similar masses of the $u$ and $d$ quarks really does seem to indicate they are "almost" the same particle, just with different specific electric charges attached.
This is important historically because the closeness of the $u$ and $d$ masses was first noted, albeit indirectly, several decades before the existence of quarks had even been postulated. Talk about premonitions!
The reason of course is that since $u$ and $d$ quarks can combine to form other particles such as neutrons and protons, those composite particles also necessarily exhibit the same similarities as the $u$ and $d$ quarks from which they are composed. The neutron and proton were in fact the very first hint of this quark link, since it was as far back as the 1930s that these two particles were oddly and unexpected similar not only in mass, but in how they interacted with each other (the strong force). The reason is now easy to explain: Since the $u$ and $d$ quarks have similar masses and in fact are pretty much the "same" particles except for their electric charges, and since both the neutron and proton were composed of sets of three up and down quarks ($n=\{u,d,d\}$ and $p=\{u,u,d\}$), of course the resulting neutrons and protons looked almost identical except for their electric charges. They were just reflecting the more fundamental link between them, the one between their constituent $u$ and $d$ quarks with their seemingly "minor" separation by exactly one unit of electric charge.
Long after the proton and neutron were discovered and characterized, new classes of particles proved to be composed out of three $u$ and $d$ quarks or their antiparticles $\overline{u}$ (charge $-\frac{2}{3}$) and $\overline{d}$ (charge $+\frac{1}{3}$). As long as the resulting two-quark mesons and three-quark hadrons are composed only of up and down quarks or anti-quarks, they continued to reflect the deep "almost the same particle" link between their $u$ and $d$ constituents.
One result of this additive process is the formation of short daisy chains of composite particles that are all very similar to each other, except for charge. The adjacent members of such daisy chains are always separated by exactly one unit of electric charge, since that is the smallest possible unit of change between the underlying $u$ and $d$ quarks. The proton and neutron are an example of the shortest possible such daisy chain, while the longest one has four members called the delta baryons. Like neutrons and protons, the delta baryons contain three $u$ or $d$ quarks. However, delta baryons have more spin than the neutron and proton, which keeps them distinct.
The elements of the delta baryon daisy chain have electric charges of $\{+2, +1, 0, -1\}$. Just by knowing that each delta baryon has three up and down quarks makes it easy to guess the composition of its first and last members: the $+2$ delta baryon has three $u$ quarks for $(3)(+\frac{2}{3})=+2$ electric charge, while the $-1$ delta baryon has three $d$ quarks for $(3)(-\frac{1}{3})=-1$ electric charge.
Alas, and unfortunately for the easy calculation of electrical charges, the way in which the electric charges are accounted by Standard Model for such daisy chains is… not simple. For one thing, it changes depending on the length of the daisy chain, and for another thing, the accounting method looks only at the magnitude of the charge changes across the entire daisy chain, not the charges of the particles.
So for example, the four ($n=4$) delta baryons span a distance of 3 charge units (that is, $n-1$) for the three "gaps" between its members, $[+2 … +1 … 0 … -1]$. The next step in this new accounting systems is to set the "zero point" smack in the center of the daisy chain, which in this case puts it halfway between the real particle charges of $+1$ and $0$. To maintain the total charge length of 3 units, the leftmost and rightmost members of the chain must be assigned values of plus or minus half the length of the chain. To get the other element values, you keep subtracting (or adding) one charge unit from each end.
Here are several examples of how the new accounting system works out for several daisy chains:
Four-Element Daisy Chain: Delta Baryons
$\begin{array}{rrrrrr}
\text{Real particle electric charges:} & & +2 & +1 & 0 & -1
\\
\text{New charge accounting system:} & & +\frac{3}{2} & +\frac{1}{2} & -\frac{1}{2} & -\frac{3}{2}
\end{array}$
Three-Element Daisy Chain: Pi Mesons
$\begin{array}{rrrrr}
\text{Real particle electric charges:} & & +1 & 0 & -1
\\
\text{New charge accounting system:} & & +1 & 0 & -1
\end{array}$
Two-Element Daisy Chain: Proton and Neutron
$\begin{array}{rrrr}
\text{Real particle electric charges:} & & +1 & 0
\\
\text{New charge accounting system:} & & +\frac{1}{2} & -\frac{1}{2}
\end{array}$
Two-Element Daisy Chain: Up Quark and Down Quark
$\begin{array}{rrrr}
\text{Real particle electric charges:} & & +\frac{2}{3} & -\frac{1}{3}
\\
\text{New charge accounting system:} & & +\frac{1}{2} & -\frac{1}{2}
\end{array}$
Notice that only for three-element daisy chains do you get a one-for-one match between the real electric charges of the actual particles and the values assigned by the new accounting system.
Perhaps the oddest of all of the accounting changes occurs in the 2-element chain comprised of the up and down quark chain. In that case you end up taking electric charges that are in $\frac{1}{3}$ increment sizes and translating them into units with $\frac{1}{2}$ increment sizes. This shifting of the quark electric charge scale by $\frac{1}{6} e$ ends up being propagated mathematically into the concept of the weak hypercharge.
Given all of that, it is likely no great surprise that the unit for this new charge accounting system makes no mention of electric charge. It is instead called isospin, which in turn is usually a shorthand way of saying the third component of isospin. Its specific value for a particle thus is usually labeled $I_3$.
More Quarks = More Innovative Accounting
As it turns out, when it comes to fermions, nature likes to do things in threes.
The down and up quarks have two families of heavier analogs. Generation II includes the strange and charm quarks, and generation III the bottom and top quarks. There do not appear to be any other generations, just those three.
What's quite odd (I dare not say "strange") about these heavier quarks is just how much they look like the down and up quarks. Their only real difference seems to be mass, which increases rapidly both across generations and within each generation.
That last phrase is critical. Unlike the down and up quarks that have almost identical masses within particles, the $+\frac{2}{3}$, up-like charmed quark is about 12 times heavier than its down-like partner, the $-\frac{1}{3}$ strange quark. The situation gets even worse for generation III, where the staggeringly massive up-like top quark is about 41 heavier than its down-like partner, the bottom quark.
Recall how I said that some pairs of fermions linked by a one-unit electric charge difference share very similar properties otherwise, such as similar masses?
Obviously then, no matter how similar the up and down quarks may seem, that similarity starts breaking down in the higher quark generations. For mass at least, it doesn't just break down, it's pretty much annihilated.
You may be wondering by now what any of this has to do with electric charge accounting?
The connection is that such considerations led to isospin being defined in an oddly narrow way that only accounts for the charges of the $u$ and $d$ quarks, not of any of the higher quarks.
The whole point of isospin is that exchanging up and down quarks results in particles with similar masses. Alas, this principle fails rather spectacularly when you start tossing in any of the higher four quarks. The strange quark is actually not too bad, but it still alters the masses of the particles significantly when it's added in. Thus instead of building simple daisy chains, you wind up building little two-dimensional arrays of particles instead.
The result was what they call "management decision" in some circles: It was decided that isospin accounting methods should apply only to the up and down quarks in particles, since only those two can be mixed and matched and still produce particles with similar masses. All of the other quarks -- the strange, charm, bottom, and top -- thus must be accounted for separately when tracking electric charge.
So ironically, the larger quarks are the ones that are tracked in a more straightforward fashion. You just count how many of them there are (anti-quarks subtract of course), and multiply that by their electric charges.
The final component of the Gell-Mann Nishijima formula is the baryon number, which amounts to looking for sets of three quarks and counting each such trio as one, regardless of the charges of the constituent quarks. If you go through the elimination of terms, the combination of the baryon number and various higher-mass quark counts compensate for the shifted and incomplete ($u$ and $d$ only) electric charge values of $I_3$.
So, if you are wondering whether it might be simpler just to add up all of the individual quark charges for a given particle, the answer is easy: yes.
Conclusion
Your final questions were:
What is the physical interpretation behind this relationship? Is charge really isospin or vice-versa? Are they manifestations of each other in the same sense as electricity and magnetism? Is it just a coincidence?
Isospin always depends on electric charge, and it can accurately be viewed as electric charge modified by a complicated set of accounting rules, rules that include determining which up-and-down-only daisy chain of particles it is in, and then resetting the electric charge zero point to the middle of that daisy chain. For that reason it seems hard not to say that isospin that is mostly just electric charge viewed through the lens of a rather remarkably complicated accounting system. Since many of the features of that system are quite arbitrary, isospin is probably more a lesson in the history of physics -- and perhaps in the dangers of opacity and over-complication of fundamentally simple ideas if you do not guard against it carefully -- than it is a feature of physics itself.
But there is nonetheless some very deep physics involved in isospin, and that is the part about why a delta of one unit of electric charge defines a profound similarity link -- a symmetry to use terminology that I've been intentionally avoiding here -- between certain pairs of particles.
Remarkably, there really are only two such pairs that exemplify the main issues, if you ignore for a moment both the echoes of them in the higher generations of quarks and the composition of their outcomes in the mesons and hadrons. Those two canonical pairs are the up and down quarks, and the electron and electron neutrino. Exploring these two pairs and how they interact gets into the issue of "weak isospin," a concept that originated in isospin but took on a life and specificity of its own. Weak isospin tells how to use the same weak force boson to transform between the particles in each pair. Weak isospin also brings the more massive quarks back into the fold, since it specifies how they too can be transformed (within limits) via weak interactions.
