The only Western Gemanic language that I know that lost the -g at the end of day is English but one would expect Finnish to have gotten their loans from a Northern Germanic language.
In early Germanic medial -g- was a fricative. It retained this valued in Norse (dagr). In Old English some inflectional forms had front vowels, so it had the value -j- in such words: daeges. This was given the spelling -y in Middle English, so day.
Modern Finnish has no G (except in NG, which is a mutation of NK). There's a softening mutation of consonants between vowels when starting a closed syllable: T becomes D, P becomes V, and K disappears (and double TT PP KK become single). So in older Finnish there would have been a fricative GH that K turned into.
My guess (and only a guess) is that the -tai of the Finnish day names is not a borrowing from a Germanic *-dai, but a development within Finnish from *-tagh borrowed from *-dagh.
I have an idea that Germanic and Finnic directly abutted in what is now Poland, before Slavs spread into the area.
