Different materials will definitely react differently - it depends on how easily they cut and how much burns away when they do. The same thickness is only a start. So yes, if you want snap-together pieces you’re going to need to adjust your settings for each material.
That being said - instead of adjusting your design, you should run tests to get the settings that do what you need need. Cut a 1" square of your MDF - and then measure it, see how much you loose - that’s your kerf. Now cut a 1" square of your plywood - measure it - and see what difference you’re talking about.
If the kerf is larger (which sounds like that’s going to be the answer) try increasing the speed or lowering the power on your cut until the kerfs match. You may find that it never does, which would mean adjusting your pattern - but this should be faster.
@evansd2’s test description may help you get an idea of the steps: