How can I determine if a mirror is tempered glass or not?
First really look for the watermarks.  These are usually ultrafaint in a corner.  They could even be upside down.  

Second look for any chips on the edges.  If there are chips it isn't tempered.

Third if you need the hole just drill it.  Use the smallest bit you have to start and work your way to the size you need.  Wear gloves but putting a tiny hole in tempered glass will probably cause a shatter but probably not the whole sheet.   There is probably a 1% chance this is tempered if that as a vanity has no code requirement for that and I don't know of builders willing to pay triple for no reason.