Indeed such problems have been studied before.  First note that if $[k_1,\ldots,k_r]$ is a given multiset (with all $k_j\ge 1$) then the count of numbers below $x$ with factorization type $p_1^{k_1}\cdots p_{r}^{k_r}$ will be at most the number of integers below $x$ with factorization type $p_1\cdots p_r$:  this is clear because if $p_1^{k_1} p_2^{k_2}\cdots p_r^{k_r} \le x$ then automatically 
$p_1p_2\cdots p_r \le x$, so that any integer counted in the first case gives rise to a corresponding one counted in the second case.  Thus the champion multisets will always be in the form of a number of $1$'s.  If you have $r$ ones, then you're asking for the number of square-free integers up to $x$ with exactly $r$ prime factors.  This has been widely investigated.  For example, [Balazard][1] establishing a conjecture of Erdős showed that for each large $x$ this sequence is unimodal in $r$ (see Theorem E on page 24), and (as one can see by Hardy-Ramanujan/Erdős-Kac) attains its maximum for $r$ close to $\log \log x$.  Balazard's paper will have more references -- in particular to work of Hildebrand and Tenenebaum giving asymptotics in wide ranges. 

[1]: http://archive.numdam.org/ARCHIVE/JTNB/JTNB_1990__2_1/JTNB_1990__2_1_13_0/JTNB_1990__2_1_13_0.pdf