I was fulling around in the CS, and I noticed that Vivec and Mournhold both have a lot of dunmer npcs in the class noble that aren't affiliated with a faction. Now, judging from my knowledge of lore, this doesn't really make sense. A noble has to be part of a Great House. So, being that both cities are religious cities, I thought it was highly likely that these NPCs are likely House Indoril. Don't know where to go with this, or if it actually matters, but just thought this was interesting and wanted to point it out.

Does a Noble have to be in a House?
I just assumed they could be very rich. There's a lot of money to be made on Vvardenfell.

Either way, just because the CS doesn't have them listed in a House doesn't mean they aren't meant to portray a member from one of the three Houses.
Also, by labeling them a House member in the CS would change all their dialog, so perhaps that's why not every Noble is a House member.

Wild guess...the land title system pretty much reflects medieval feudalism, so anyone not a member of a house is some sort of tenant and tenants are at some level basically property. I don't think any amount of wealth could make someone a noble, where even the 'poor' house members have some share of the house's land title so house rank could confer nobility. Compare to modern Saudi Arabia...there are plenty of (incredibly) wealthy people, but they are still subjects of the House of Saud...which at somewhere around 30,000 family members no doubt includes some pretty downtrodden 'royalty'.

As to the initial question, I'd be willing to accept the idea that they are members of either Indoril or Dres who are in Vivec to interact with the temple...short term passing through or long term making their home in temple managed quarters. One would think members of the three houses that have their own canton would be designated appropriately and probably live there.

Pure speculation, but if it holds together of a piece and works for your play, why not?

You don't have to be part of a Great House to be a noble in Morrowind. Respected members of minor houses can be nobles as well. And I'd assume it'd be possible for outlanders to rise to nobility in minor houses just like in the Great Houses.

And from what I know, being rich doesn't "legally" make you a noble or lord/lady. Those individuals are usually called things like merchant princes.