Do you think certain LIs have a "type"?

You mean other than Sera who openly declared an attraction to Qunari women? There was Kaidan who did say he preferred more adventurous women.

Or did you mean a type of personality?

Then I'd say someone like Blackwall might prefer someone with a sense of justice, but also mercy, while Josephine may prefer someone who respects others as she's mentioned deploring rudeness from those who know better. Sera obviously would want to hang around someone with enough of a sense of humor to have fun with.

Sera: Tall human women, someone with a big torso and arms that she can feel safe in. Someone who has lived a hard life, and is humble and open-minded enough to speak to her in ways that don't make her feel stupid or crazy or childish.

Cassandra: Someone decisive who makes her feel like she isn't the only one taking responsibility and willing to shoulder the heavy loads, and who doesn't seem surprised every time she shows vulnerability, as if they expect her to act like a big hunk of iron all the time.

Josephine: An actual citizen of Thedas with family and ties to a community and a place in the world, who she can bring home to meet her family without feeling like she's dragging something in from the gutter, and who values the discipline and thoughtfulness she brings to the table, and doesn't make her feel a need to defend her purpose there just because she isn't a ruthless killer.

Dorian: Someone who has broken their mold and decided their own values based on sense and compassion and experience, and actually stuck to them. He doesn't strike me as someone who would respect a mouthpiece of their community or institution or any established philosophical doctrine, he's just too skeptical by nature, but at the same time he's very civic-minded and doesn't approve of anarchistic or antisocial notions. You have to want to do right by people, even if not in the particular ways they imagine you should.

Iron Bull: No real type. I think he's too messed up by Qunari culture's approach to love and the stresses of his lifestyle to have a real concept of or desire for relationships as we understand them. Probably just someone willing to mess around who doesn't ask a lot of questions, forcing him to spend energy lying and dissembling.

Solas: An Elven woman who unequivocally is on the side of her people and would defend them to the death, and who is willing to make tough sacrifices on their behalf, but is still intelligent and open-minded enough to learn about the world and give people and concepts she doesn't fully understand a chance to show their merits. Someone who makes him feel that the strength and heart and grace of the ancient elves that he fought so bitterly to preserve from injustice haven't been completely smothered by his actions, even though that knowledge is bittersweet at best. Like Cassandra, he dearly misses someone to share his load even though he knows it's impossible.

Cullen(DAI): Someone who admires his courage and discipline, and wants to support him in his weaker moments and restrain him in his stupider ones. He knows he isn't the smartest or wisest person in any given room, and feels better knowing that someone else who at least understands and respects his concerns is there to help make the best decision, making him more comfortable focusing on his military duties and simply being a strong weapon in the hands of the righteous the way he was raised and trained to.

An interesting twist on the shining knight archetype in Templars is that they really are conditioned to take orders and instruction and guidance from unarmed and unarmored women. I doubt he could wean himself off that as efficiently as he does Lyrium.   

No particular thoughts on Blackwall. His themes of finding purpose and redemption in responsibility and trying to inspire strength rather than shelter weakness are pretty powerful, but the character himself doesn't extend very far beyond them and isn't very consistently written to my mind.