From Proto-Indo-European *dṓm, from root *demh₂- (“to build”). Cognates include Ancient Greek δόμος (dómos), Albanian dhomë (“a chamber, a room”), Sanskrit दम (dáma) and Proto-Slavic *domъ. The same Proto-Indo-European root also gave Old English timber (“building, act of building”) (English timber).
From Proto-Indo-European *dem-h₂- (“to domesticate, tame”). Cognate with Sanskrit दाम्यति (dāmyati), Ancient Greek δαμνάω (damnaō), Old High German zemmen and the Proto-Germanic adjective *tamaz.
The following sentence makes me wonder and have this question: tame is a reflex of PIE *dem-h₂- (“to domesticate, tame”), which has the same formation of PIE *demh₂- (“to build”). Maybe the semantic explanation is "tamed in a house"?
Old English tom, tam "domesticated, docile," from Proto-Germanic *tamaz (cf. Old Norse tamr, Old Saxon, Old Frisian, Middle Low German, Middle Dutch tam, Old High German zam, German zahm "tame," Gothic tamjan "to tame"), from PIE *deme- "to constrain, to force, to break (horses)" (cf. Sanskrit damayati "tames;" Persian dam "a tame animal;" Greek daman "to tame, subdue," dmetos "tame;" Latin domare "to tame, subdue;" Old Irish damnaim "I tie up, fasten, I tame, subdue"). Possible ulterior connection with PIE *dem- "house, household" (see domestic). Meaning "spiritless, weak, dull" is recorded from c.1600.