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Even though mass and charge are two different physical properties of matter, have there been any attempts at unifying them? say, by defining a physical property of matter such that mass and charge are naturally derived from it, and are regarded as different manifestations of the same physical phenomenon?

Kind of like how the electric and magnetic fields are related by Maxwell's equations (also the fact that a magnetic field is basically an electric field in a different reference frame), or how mass and energy are related.

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Yes, there have been attempts to unify these concepts. The earliest was done by Kaluza and Klein, who imagined that we lived in a $5$ dimensional universe, and the electromagnetic field was really a manifestation of gravity in the extra, fifth dimension.

Gravity, which is a manifestation of the curvature of spacetime, is described by the metric of spacetime $g_{\mu\nu}$, a $4\times 4$ matrix. In this five dimensional universe, we have a five dimensional metric $\tilde{g}_{ab}$, a $5\times 5$ matrix, parametrized by

$$\tilde{g}_{ab}=\begin{pmatrix}g_{\mu\nu}+\phi^2A_{\mu}A_{\nu} && \phi^2 A_{\mu}\\ \phi^2 A_{\nu} && \phi^2 \end{pmatrix}$$

Where $g_{\mu\nu}$ is a four dimensional metric, $A_{\mu}$ is a four dimensional vector potential, which describes the electromagnetic field, and a scalar field $\phi$. If $\phi$ is constant, the five dimensional Einstein field equations

$$\tilde{R}_{ab}-\frac{1}{2}\tilde{g}_{ab}\tilde{R}=0$$

miraculously reproduce both the four dimensional Einstein field equations, and Maxwell's equations.

And so in this five dimensional universe, charge is just a different manifestation of the five dimensional energy-momentum tensor $\tilde{T}_{ab}$ that would appear on the RHS of the above equation.

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    $\begingroup$ It is perhaps worth mentioning that the relationship between mass and charge which results from Kaluza-Klein theory is not what we observe in experiments, which is why the theory is not considered to be an accurate model. $\endgroup$
    – J. Murray
    Commented Feb 6, 2022 at 21:43
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This is maybe not exactly what you're asking for, but there is a direct relationship between charge and mass.

Namely, there exists a canonical isomorphism between the physical dimensions of mass and charge which is induced by Newton's gravitational law and Coulomb's law: the electric charge $q$ corresponding to a mass $m$ is such that the (absolute value of) the gravitational force between two particles of mass m equals the electric force between two particles of charge $q$, for equal distances.

Note that this has nothing to do with a choice of units. However if one uses a system of units where charge and mass are measured in the same unit, then

$$G = \frac{1}{4\pi\varepsilon_0}$$

holds.

Note: When I speak of an isomorphism of physical dimensions I mean that a physical dimension is interpreted as an $\mathbf{R}_{>0}$-torsor (a unit of a physical dimension being an element of that torsor), and an isomorphism of physical dimensions therefore means isomorphism of $\mathbf{R}_{>0}$-torsors.

More precisely, you would want to restrict to isomorphisms which are natural in the sense that the definition doesn't make reference to any particular element of that torsors (unit).

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