Yes, multinomial regression is still usable here, though you need to interpret the results accordingly. Perhaps you want to use ID to generate probabilities of different outcomes? If that's the case, then you actually must have numerous observations per ID, and the observations for each ID must have at least some variation in the outcome variable or you'll have perfect prediction for at least one of the IDs and the algorithm will lock up. Speaking of which, if that's your intention, you're going to construct dummy variables out of all the IDs, and omit one as your base variable. Or maybe you don't actually want to use ID in your model at all, and you're just worried that these observations don't all come from totally different entities (people, etc.), in which case you generally don't need to worry. You may still want to do a fixed-effects model, however, including those ID dummy variables, if, again, you have enough observations and variance, and have reason to think that constant unit-level properties/effects may bias your key inferences and theoretical conclusions. Again, it depends on your research question and the interpretation you're trying to arrive at, but there's nothing inherently wrong or inappropriate about your model choice.
(And re: the above comment, I wouldn't necessarily dub "Weekly, quarterly, annually" ordinal. They're ordinal in a certain sense, but everything is ordinal in a certain sense, and the same things are ordinal with different rankings in different ways - those three words can be ranked in terms of the number of letters they have, too. Yet again, it depends on your research question: is it relevant that weeks are shorter than quarters which are shorter than years? It might not be at all. And on top of that, treating something as ordinal involves assumptions that are simply not made when you treat it as categorical, so the worst you can do if you ignore relevant ordinality (or even continuity) is lose efficiency.)