Great question. First of all, you're absolutely right that until we find a universe with a different number of dimensions in the lab, there's no single "right" way to generalize the laws of physics to different numbers of dimensions - we need to be guided by physical intuition or philosophical preference.
But there are solid theoretical reasons for choosing to generalize E&M to different numbers of dimensions by choosing to hold Maxwell's equations "fixed" across dimensions, rather than, say, Coulomb's law, the Biot-Savart law, and the Lorentz force law. For one thing, it's hard to fit magnetism into other numbers of dimensions while keeping it as a vector field - the defining equations of 3D magnetism, the Lorentz force law and the Biot-Savart law, both involve cross products of vectors, and cross products can only be formulated in three dimensions (and also seven, but that's a weird technicality and the 7D cross product isn't as mathematically nice as the 3D one).
For another thing, a key theoretical feature of 3D E&M is that it is Lorentz-invariant and therefore compatible with special relativity, so we'd like to keep that true in other numbers of dimensions. And the relativistically covariant form of E&M much more directly reduces to Maxwell's equations in a given Lorentz frame than to Coulomb's law.
For a third thing, 3D E&M possess a gauge symmetry and can be formulated in terms of the magnetic vector potential (these turn out to be very closely related statements). If we want to keep this true in other numbers of dimensions, then we need to use Maxwell's equations rather than Coulomb's law.
These reasons are all variations on the basic idea that if we transplanted Coulomb's law into other numbers of dimensions, then a whole bunch of really nice mathematical structure that the 3D version possesses would immediately fall apart.
Best Answer
Relativity treats spacetime as a four dimensional manifold equipped with a metric. We can choose any system of coordinates we want to measure out the spacetime. It's natural for us humans to choose something like $(t, x, y, z)$, but this is not the only choice. Even in special relativity the Lorentz transformations mix up the time and spatial coordinates, so what looks like a purely time displacement to us may look like a mixed time and space displacement to a moving observer. In general relativity we may choose coordinate systems like Kruskal-Szekeres coordinates where there is no time coordinate in the sense we usually understand the term.
So in this sense there is a symmetry between space and time coordinates because there is no unique separation between space and time. This is the point Harold makes in the comments.
But this does not answer your question. Regardless of what coordinate system we use, locally the metric will always look like:
$$ ds^2 = -da^2 + db^2 + dc^2 + dd^2 \tag{1} $$
where I've deliberately used arbitrary coordinates $(a, b, c, d)$ to avoid selecting one of them as time. If you look at equation (1) it should immediately strike you that there are three + signs and one - sign, so there is a fundamental asymmetry. It's the dimension with the - sign that is the timelike dimension and the ones with the + sign are spacelike. Whatever coordinates we choose we always find that there is are three spacelike dimensions and one timelike. We call this the signature of the spacetime and write it as $(-+++)$ or if you prefer $(+---)$.
Which brings us back to your question why are there three spacelike dimensions and one timelike dimension? And there is no answer to this because General Relativity contains no symmetry principle to specify what the signature is. If you decided you wanted two or three timelike dimensions you could still plug these into Einstein's equation and do the maths. The problem is that with more than one timelike dimension the equations become ultrahyperbolic and cannot describe a universe like the one we see around us.
So the only answer I can give to your question is that there is only one time dimension because if there were more time dimensions we wouldn't be here to see them.