Both these words come from Old English. They were written as “hus” and “mus” back then, and the plurals were “hus” and “mys”. That is, the plural for “hus” was the same as singular, while the plural for “mus” changed the vowel. This difference in plurals is due to the fact that Old English was a gendered language, and while “mus” was a feminine noun (basically like we call a ship “she” today), “hus” was neuter. Nouns of different gender had different plural formation rules. Eventually English invented the new plural formation rule (by adding -s at the end of a word), but didn’t apply it to some feminine nouns like “mus”/“mouse” which retained their old plurals. Then English nouns lost their gender, and the vowels in words changed rather uniformly during the linguistic event called “The Great Vowel Shift”.
Random fun fact about the English word mouse: The origins of that particular word can be traced back to an original Proto-Indo-European word that was spoken between 4500 and 6500 years ago. It was probably pronounced something like mus, with the vowel pronunciation a little unclear - it's changed in remarkably little in English from its hypothesized original form. From this ancient language descended many hundreds of other languages including English, Greek, Latin, Persian, and Hindi. It's also from this original word that the Latin speaking physicians came up with the term muscle because our muscles looked like little mice when contracted. Then that got imported into English at some point. The Greek prefix myo (as in Myocarditis) also comes from the same roots.
There's a lot of words that originate from a common word ancestor like that. It's really interesting.