Old English attests the word as docga. My hypothesis is that it’s dōc “bastard, mongrel” + -ga [diminutive suffix], roughly like calling it “the little mutt”. The vowel shortening would’ve been happened already back then, otherwise the modern form would be *doog /du:g/.
Note that Latin/Romance attests similar phenomena (depreciative word for animal becoming the default word + diminutives being ingrained into the main word). And typically when you see weird stuff going on in a language you’ll see it happening in other languages too.
Who on earth calls their feet “dogs”? I’ve also never heard of “putting on the dog”, but that’s probably just because. 🐶
Calling feet dogs is outdated, but used to be pretty common, especially saying “my dogs are tired.” They even reference it in old cartoons. A character will take off their shoes and their feet will turn into barking dogs.
