The action for the free Maxwell theory is given by $$S=\int d^dx\sqrt{-g}\bigg(-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}\bigg)$$ The theory is invariant under conformal transformations $g_{\mu\nu}\to\Omega^2(x)g_{\mu\nu}$ only in $d=4$ as can be recognized by looking at the trace of the energy-momentum tensor of the theory, or more directly by recognizing that under such a transformation,
- $F^{\mu\nu}F_{\mu\nu}=F_{\mu\nu}F_{\alpha\beta}g^{\alpha\mu}g^{\beta\nu}\to \Omega^{-4}F_{\mu\nu}F_{\alpha\beta}g^{\alpha\mu}g^{\beta\nu}=\Omega^{-4}F^{\mu\nu}F_{\mu\nu}$
- $g=e^{\text{Tr}(\ln(g_{\mu\nu}))}\to e^{\text{Tr}(\ln(\Omega^2g_{\mu\nu}))}=e^{\text{Tr}(2\ln(\Omega))}g=e^{2d\ln(\Omega)}g=\Omega^{2d}g$
and thus, for $-\frac{1}{4}\sqrt{-g}F^{\mu\nu}F_{\mu\nu}$ to be invariant, $\frac{\Omega^{d}}{\Omega^4}=1$ which is the case only in $d=4$.
This means that the free Maxwell theory is not conformally invariant except in $d=4$. However, the definition of theory is the same in all dimensions and doesn't involve any dimensionful parameter, so I am confused as to what sets the scale of the problem in $d\neq 4$ when the theory is not conformally invariant.
