They have the same meaning, yes, as you seemed to know already.

If any, differences between the two will be minimal in general use. *A la mode* will maybe be slightly more employed when actually refering to (clothes) fashion, but both variants are equivalent for other contexts.

For your second question, no, *à la mode* isn't used in this sense for ice cream on desserts. I think it's another case of the *belgian waffle* (unknown as such in Belgium) / *crême anglaise* (unknown as such in England) / *french fries* (unknown as such in France) / ...