There's not really a precedent set for this in canon, as it's not really touched on or shown to exist much - 'marriage' in heaven is a different matter than it is for most humans.

As a rule, gods are probably careful about anything that might compromise their shinki's loyalty to them and the bond a shinki has to their god is considered The Most Important One. That said, excuses could be made for it - especially if a god's permission is granted for such a relationship. Serving in the same temple would probably cause much less strife for one of these arrangements as well.

Basically, it would be understandable - if weird, given their situations - if some shinki cling to some of the more human traditions, but the acceptance of it would probably differ between gods and temples and on a case-by-case basis. Ultimately if an old god wanted to allow it, for any reason, that was seen as their prerogative, but it would not be considered a common pratice for shinki.

As for what kinds of relationships and how accepted/common they could be considered among the old gods and their shinki:
Uncommon (Shinki-Shinki Intra-Temple)
Rare (God-God, Shinki-Shinki Cross-Temple)
Ultra-Rare / Taboo / Nora levels of WTF Are You Doing (God-Shinki)