Is it okay to inquire about the status of a paper when the online submission system shows no update three months after submission? Around 3 months ago, I submitted a paper to a Math journal for peer-review through an online submission system. The same day I received in my e-mail an acknowledgement saying that the paper had been received.

The problem is that the status of the paper on the online submission system is still "Manuscript submitted".

I think this means that the paper has not even been assigned to an editor yet (once this happens, the editor will probably start looking for reviewers), but as far as I know submissions are assigned to editors within a few weeks (I once submitted a paper to another journal and when an editor had been assigned, the status on the online system changed to "Editor handles" or something like that).

I know that publishing is a long and complicated process and I'm not trying to rush anybody, but sometimes manuscripts are lost in the system. In fact, this has once happened to me, so I just want to make sure that the paper was not lost in the system.

Should I send an e-mail asking for the status of the paper?

PS: I think that, at the time of submitting the paper, I didn't choose any specific editor to submit the paper to.
Yes, you should send a quick email asking for an update. In my experience, every electronic manuscript submission/tracking system seem to be different. Even different journals from the same publisher who use a unified log in system, seem to have tweaks to what you see. I think that after one month at any one stage, with the exception of "under review", it is perfectly reasonable, and probably desirable, to ask for a status update. The time a manuscript spend "under review" is often very field, and even journal, specific and can range from days to 12+ months.

When sending an email, be polite and explain that you just want to make sure that you do not need to do anything at the current stage. In all likelihood the status update will be no more informative than "we are working on it" (i.e., they haven't lost it), but that is informative.