When or why to use the $\equiv$ symbol in place of the $=$ symbol? In literature, I read the following:

A typical relationship*, often appearing in the literature, is:
$$|-\nabla(\bar p+\rho g z)|\equiv \rho g J=q(\mu w+\rho Bq^m)$$

The nomenclature does not define the $\equiv$ symbol. I think it means "equivalent to" or "defined as". If it does mean that, what is its purpose? That is, what information would be lost (or error incurred) if the relationship* was written with an "equal to" sign $=$ in place of the "equivalent to" sign $\equiv$? E.g.,:
$$|-\nabla(\bar p+\rho g z)| = \rho g J=q(\mu w+\rho Bq^m)$$
*A secondary question: Is the use of the term "relationship" specific to this situation? I.e., can we not refer to the above as an "equation"?
 A: Like $:=$ or $\overset{\text{def}}{=}$, $\equiv$ is indeed a moderately common way to distinguish a definition of the l.h.s. in terms of the r.h.s. from any other sort of equality relation ("equation", "relationship", "relation" are all completely interchangeable here).
The purpose is to clearly distinguish between equations, which can/need to be derived, and definitions, which are simply stated and need no proof. There are plenty of texts that do not use special notation for definitions, so this is clearly not universally considered to be necessary.
A: Different authors have different styles and conventions.  My personal convention goes as follows:

*

*I use the $\equiv$ symbol essentially to denote shorthand, or something which is trivially true by definition of notation. There is no logical or physical content here; two expressions which are separated by $\equiv$ have exactly the same meaning, but are written a different way.

*I use the $=$ symbol to denote the equality of two expressions which are not the same by definition. This could correspond to a true but non-obvious relationship, or a constraint that I am imposing.

*I use the $:=$ symbol to denote a concrete assignment, like a function definition.


As an example which distinguishes between the first two symbols, consider this expression of Gauss's law:
$$\Phi_E \equiv \oint \mathbf E \cdot \mathrm d\mathbf S = \frac{Q}{\epsilon_0}$$
There is no content in the $\equiv$ symbol; when I write $\Phi_E$ on a piece of paper, what I mean is $\oint \mathbf E \cdot \mathrm d\mathbf S$.  On the other hand, there is content in the $=$ symbol.
As another example, consider the following lines:
$$\mathbf F_{net} \equiv \sum_i \mathbf F_i = m\mathbf a$$
$$\mathbf F_{net} := -mg \hat z \implies \mathbf a = -g \hat z$$

This convention of mine is non-standard.  I use it when I feel like it's appropriate and would clarify rather than add confusion. I personally think it's important (and pedagogically useful) to distinguish between things which are true by virtue of notation, things which are not defined to be the same but are nonetheless equal, and an assignment of value, but my opinion is not universally shared.
Your author is doing something similar, though apparently not using precisely the same notation.

A secondary question: Is the use of the term "relationship" specific to this situation? I.e., can we not refer to the above as an "equation"?

You could, if you wanted to. To me, replacing the word "relationship" with "equation" feels ... clunky, somehow. It sounds like a sentence a primary school student would use, not unlike referring to $\pi = 3.14159\ldots$ as "the equation for $\pi$."
Your author is giving an example of a relationship between some physical quantities. That relationship takes the form of an equality between two expressions. Whether you'd like to refer to it as a relationship or as an equation is ultimately a stylistic choice, I think.
A: Also, although the same symbol, $=$, is usually used to represent assignment and equality, in many programming languages different symbols are used.  For example, Java uses $==$ for the equality operation and $=$ for the assignment operation.  The equality operation checks if two variables have the same value.  The assignment operation assigns the value to be stored at the address on the left side to the value on the right side. So, for example, 5 = x is an invalid statement because 5 is not an address; x == 5 is valid. 5 == x is true if the value (stored at the address for variable x) is 5.
(In Java, $==$ compares values for primitive types. In Java, for objects, $==$ compares addresses, not values; $equals()$ compares values. Pascal uses $:=$ for assignment and $=$ for equality.)
