Regarding the electromagnetic field, a fundamental invariant is $E^2-B^2$. $E$ is the electric field, $B$ the magnetic field.
Regarding the four-momentum of a particle, a fundamental invariant is $\mathscr{E}^2-p^2$. Here, $\mathscr{E}$ is the total energy, $p$ the momentum.
Is the similarity of both equations coming from some mathematically equivalent structure (which?) or is it pure chance?
Yes, there is a reason that the form of the invariants is a mixed sign quadratic form. It comes from the fact that in both cases, the quantities involved are components of a Lorentz tensor, and both expressions are contractions of a Lorentz tensor with itself and with the Minkowski tensor.
Any Lorentz tensor defines an invariant when it is contracted with itself and with the Minkowski tensor, which has components $$ \eta_{\mu\nu} = \begin{bmatrix} -1~~~ & \!\!\! 0 & 0 & 0~~ \\ 0 & \!\!\!1 & 0 & 0~~ \\ 0 & \!\!\!0 & 1 & 0~~ \\ 0 & \!\!\!0 & 0 & 1~~ \end{bmatrix}, $$ the latter used the appropriate number of times to get a single number. Notice one entry is negative; this causes the invariant quadratic form to have a negative square term in it.
$\mathscr{E},p_1,p_2,p_3$ are components $p^\mu$ of the tensor of energy-momentum of a particle, and $E_1,E_2,E_3,B_1,B_2,B_3$ are components $F^{\mu\nu}$ of the Faraday tensor of EM field.
The contraction definining an invariant number looks like this for rank 1 tensors:
$$ C_1 = \eta_{\rho\sigma} p^{\sigma} p^{\rho} = p_\rho p^\rho, $$ giving $C_1 = -\mathscr{E}^2/c^2 + p^2$, and like this for rank 2 tensors ($\eta$ is used twice):
$$ C_2 = \eta_{\rho\alpha}\eta_{\sigma\beta} F^{\rho\sigma} F^{\alpha\beta} = F_{\alpha\beta} F^{\alpha\beta}, $$ giving $C_2 = -2(E_1^2 + E_2^2 + E_3^2)/c^2 + 2(B_1^2 + B_2^2 + B_3^2)$ .
Firstly, the second equation $\mathscr{E}^2-p^2$ is not related to gravity, it merely states that the four vector $p^\mu$ has the same rest mass in every system. (note that $\mathscr{E}$ here is energy)
Secondly, $B$ and $E$ are certainly related in relativity, however, unlike the case mentioned above, $B$ and $E$ (this one is electric field) is not a set of four vectors (because $B$ and $E$ themselves are vectors), they have a slightly more complicated relation $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$, where the $F_{\mu\nu}$, when labeled with different indices, represent magnetic field and electric field in different direction.
In short, they have certain similarity, but they are not exactly equivalent.
