I personally don't think sexuality has any impact on a person's personality. The love interests would be just as defined as characters in there personalities regardless of what sexualities they had, even if they're all bisexual.
If we're going to start imposing limits to help define the characters, gender isn't the place to start. Values and ideals are the place to start. Cassandra and Cullen shouldn't romance a blood mage, Solas and Sera shouldn't romance people who are bigoted against elves, Iron Bull shouldn't romance someone who's bigoted against qunari, Vivienne, Dorian, and Solas shouldn't romance someone who's bigoted against mages.
Those are the distinctions that matter in regards to the integrity of their characters, in my opinion.
Eh, I think sexuality can impact your personality to some degree, just because the things you experience impact your personality. Being bisexual has definitely had at least some of an impact on my personality, since I grew up in a community where bisexual people weren't even acknowledged and gay people were only stereotypes on pride parades. (Even now, my parents--who do love me very much--believe that my same sex attraction is a symptom of a disease and calling myself a bisexual is somehow making that my ENTIRE identity.) I'd like to think that my experiences with struggling with my sexuality has made me at least a little bit more empathetic to people who've felt displaced in their communities--and it's CERTAINLY impacted how I feel about my faith, both in a positive way (since, without getting too preachy, a big part of my faith is believing in a God who loves me for who I am and doesn't see me as "less than" straight people, which has helped a lot in accepting myself--not just as a bisexual person but also as someone that's maybe not "exactly who I'd like to be") and in a negative way (in the sense that I'm very distrustful of the dominant, conservative Christian culture in the USA).
Of course, that's the world we live in. In a more ideal world where sexuality isn't as much of an issue--which Thedas seems to be (well, not ideal, but not involved in a "culture war" surrounding sexuality)--it probably wouldn't have that huge of an impact.
Also I'd love the idea of, say, not being able to romance a character if you had opinions that were a deal breaker to them. (Although I also kinda liked the rivalry/friendship system in 2 where part of the romance was sometimes the character struggling with the idea that they loved you DESPITE having such a huge difference of opinion on a subject that was important to them.)