Tagging the number of entrances to apartment buildings

First of all, this degree of micro detail is not widespread yet. You will be counting the amount of entrance= added in OSM only. Better find your country's open data, if they have this info at all on buildings.

This seems difficult to define. What about private, service, and emergency ones? Then there's the problem of internal doors to a courtyard, so you would need think about building doors vs site access points (which may not be a door, or have any gates) to public streets. An access point could further have multiple sides, with individual doors added at the extreme end. If there is a external staircase, above-ground access, or even individual doors on each floors, you will effectively be double counting many. Computing this will be expensive, or not possible entirely from the lack of tags.

Could be more narrow yet specific to count addr:flats= (interestingly only half are used together with entrance=), and addr:unit if it is used according to the original definition of different parts accessible by a certain door only (instead of the common meaning of "unit" that's akin to addr:door=). Address data is expected to be more complete in OSM, although the extent is more on addr:housenumber= depending on if there is an import. Then you will at least have some idea of how they are distributed.