Both Relanna and Amarel collected the same things: keys, books and letters, and unique (preferably enchanted) items.
With Relanna, I could justify the collecting unique items as a show of power. Having a host of Daedric artifacts, along with the Staff of Worms,
and various personal gifts from the Counts and Countesses of Cyrodiil made quite the impression on her guests. I'm not sure why Amarel hangs onto those things, though, since she's not the showy type. In some cases they're genuinely useful, so either she or her companions keep them close by rather than putting them on display. A few more have sentimental value: the Kvatch cuirass, for example.
As for books...well, I tend to amass books in RL, so I do it in games too.
Both characters enjoy reading and the pursuit of knowledge, although in Relanna's case sometimes it turned into a show of power, again, to have certain rare or unique books out on display.
I have no idea why I collect keys. Must be my OCD trying to kick in. Either that or too much time spent playing old school FPSes as a kid.