Is there a way to prove that one sieve will have a bigger remaining set than another? Like if we have a sieve that sieves to get all mersenne primes and a sieve that sieves to find all twin primes for instance?

Generally, it is very difficult to establish lower bounds for thin sequences via sieve theory. So while Brun's theorem (an upper bound for twin primes) was proved almost 100 years ago, we still can't prove any infinite lower bound for twin primes. Indeed, modern upper bounds for the twin primes are only off by a constant factor from what we believe to be true.

Sieving for Mersenne primes is much harder than sieving for twin primes because the structure of the divisors is more complex. But even here the same general rule holds: upper bounds are easier than lower bounds.