Tally can certainly count the occurrances of eigenvalues in your list called vals. For instance Tally[vals] tells me that there are 2 instances of $10 \pi^{2}$ and 1 instance of $5 \pi^{2}$.
Looking at Tally in the documentation I see that the word "multiplicities" means the number of instances of each distinct element in a list.
I notice that your code is producing a different vals in my notebook! Please quit your kernel and re-evaluate!
Also Wolfram mentions this in his documentation.
