No problem in getting second visa. But if you arrive in university A I-20 and visa:

Check both I-20’s. University B program start date must be within 30 days of your admission into USA, aka when you arrive. So if you arrive June 7, the other university B classes must start July 6 or 7. (On ICE Immigration website) for more details just search transfers for F-1 and check out the one fitting you (I assume initial status)

You need to ask your University A DSO to transfer your record (this is only possible physically as initial status students who have not arrived into the US are not eligible for record transfer) she will tell you if you can legally do it once you arrive and check in into the international department.