Can a character have on two identical rings, one in each ring slot? I can't seem to find any rule saying this isn't possible, but the DM thinks it may not be. The ring I have in mind is just a pair of +1 protection rings.
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You can wear them, but their effects might not stack.
In this case, two rings of protection won't stack, because the rings provide a deflection bonus to AC, but deflection bonuses don't stack with other deflection bonuses.
There is no reason you can't but it won't be any better than using just one as like bonuses don't stack:
From the d20pfsrd:
So you'd only get a +1 bonus regardless of whether you are wearing 1 or 2 rings.
As the above says about two like bonuses, however there is an advantage to having two of the same ring if they have charges or some other use limit. So I wouldn't mind having two rings of Fireball, each with a 1/day use limit or some-such.