Yes - Wirsing has shown that the number of odd perfect numbers in $[1,n]$ is at most $n^{O(1/\log\log n)}$. By Euler's correspondence between even perfect numbers and Mersenne primes it is certainly true that the number of even perfect numbers in $[1,n]$ is $\ll \log n$.
Therefore $\lvert P\cap [1,n]\rvert \leq n^{O(1/\log\log n)}$ (and in particular $P$ has zero density).
Wirsing's bound is the best known for odd perfect numbers, and hence it is unknown whether e.g. $\mu(\log P)=0$.
Furthermore, since conjecturally there are $\gg \log\log n$ many Mersenne primes in $[1,n]$, it should in fact be true that $\mu(\log\log P)>0$.