Is there an "evolutionary species similarity calculator"?
Since the simple value of percentage of nucleic acid identity over whole genomes isn't very useful in taxonomy, you won't find these values published in a database, even if they were computed by someone in the process of other work.

One project that provides ready-for-consumption phylogenetic trees based on similarity metrics, by using orthologous genes between species, is OMA. They compute all orthologous genes between two species, and they are now at 1,235 species:

http://www.biorecipes.com/Orthologues/status.html

You could then read off the "distance" between cat and mouse from the eukaryotes tree. This value would then be more meaningful than the nucleic acid identity percentage.

Apart from the trees, using their OMAbrowser you can get for any species pair the number of common orthology groups, i.e., the number of common genes between those two species.