Generally, I feel religious music and hymns would be the most present in terms of music in government and officialized settings. As for bards and such, I would assume a very natively influenced sound. We can make some inferences by the type of instruments the natives seem to have and comparing that to real life to get an estimate on that sound. They seem to really like percussive and woodwind instruments, and have a tendency to use drums and wooden windchimes a lot, excessively on the drums. All this together with the designs of the instruments, as well as other similarities between native Dunmer and IRL, is giving a very middle-eastern/ Southern Asian direction. Generally this points the sound to being something of a mix between old Hebrew music and traditional Indian music, so it would not surprise me at all if the Dunmer had something akin to a Sitar in their instrument range (I'd love it if we got to make up some fictional instruments for the region in good ol' Dwarf Fortress fashion). As for music from outside of morrowind, as it is being colonized by the empire, I think it is safe to assume it is very much akin to ancient mediterranian in influence, so I feel they'd have harps and flutes with that mediterranian flare in playing. As for the songs, probably epics, poems, and folk songs alongside hymns and chants. The Ashlanders I feel would especially be fans of spiritual chants and epics.

chitin drums is something I think would be extremely likely for the culture. Cultures like the Dunmer IRL, in my experience, LOVE drums and percussive instruments alongside some stringed ones, and tend to also like throat singing. One culture that is particularly similar would also be the Mongols in the age of the great Khans. Here is an example: https://www.youtube.com/watch?v=1rmo3fKeveo

Now as for the religious views of music, I think it wouldn't effect it too much outside of their preference of beat. More settled, high class types more in line with the temples would like probably more ordered and steady music akin to your standard hymns, while lesser folk would like more active sounds, these sounds however would still be leapyears more steady and calm in nature than, say, rock and metal; the sounds one associates with those genres in terms of beat would be more in the very uncivilized areas, such as the tribes and outer reaches (one must remember that Blues, the grandaddy of all rock, got it from slaves who had been imported from tribal areas and the like in Africa, which would then pass on to further generations and change to be more "american" as one would say, until it became the Blues we all know and love.)