I have heard multiple people say that when wiring transformers together, it is important that they are identical, with the same voltage rating, wire gauge, number of turns, and core size, but I have not been able to find an explanation as to why this is. I have found answers for why you can't combine non-identical transformers in parallel but I can't seem to find an answer for why you can't combine them in series. Specifically, if you wired two transformers in series with identical primaries but different numbers of secondary windings (same wire gauge, different sized cores). What would the ramifications be? Is there any possible way to address these problems so as to make it feasible?
Your hunch is correct. There is generally not a problem with your approach.
Figure 1. A series transformer example.
Here we have two transformers with their primaries wired in parallel and their secondaries in series. The voltages simply add and the current will be the same through both. As you reasoned, the maximum current will be determined by the transformer with the lower secondary current rating. The wise man might add a fuse to the secondary and rate this to protect the lower current winding.
Figure 2. Dropping or boosting mains voltage.
The circuit of Figure 2 is an example of a voltage reducer/booster setup. Let's say you want to reduce your mains voltage from 230 V to 200 V then you would use a 230:30 transformer and connect the secondary in anti-phase with the mains so that the output voltage subtracts. By reversing the secondary connections you would boost the output to 260 V. Note that this is relevant to your question in that there is another transformer not shown in the schematic - the utility transformer which will likely have a much higher rating than your reducer / booster.
The Figure 2 transformer will be much more compact than a full 200 V transformer. If, for the sake of easy maths, the required output current is 1 A then XFMR1 needs to be rated at 30 V × 1 A = 30 VA. If we were to choose a 230 V to 200 V transformer then its rating would be 200 V × 1 A = 200 VA.