Liam’s general thoughts on the main companions:
ED-E: A precious, small robot whom Liam is too emotionally attached to. From the time Liam fixes ED-E up in Primm, he rarely goes anywhere without the eyebot shadowing him. He cries after he has to sacrifice ED-E in the Divide to stop the missiles from launching; when he gets back to the Mojave and sees that his ED-E is just fine, he hugs him for a solid minute.
When Liam conducts official Vegas business post-game, ED-E is often floating near him, conveniently recording the conversation. (And he wouldn’t dream of going to the Commonwealth without bringing ED-E along.)
Veronica: Reminds Liam of his sister, which more or less guarantees that he adores her. Her struggles with the Brotherhood’s elders remind him a lot of his own attempts to persuade Shi-Town’s isolationists. In-game, she’s the first companion other than ED-E that he joins up with, so he’s known her longer than any of the others.
He does I Could Make You Care after House orders him to destroy the Brotherhood, because he believes he can find something that will persuade House to let the Brotherhood live. It... doesn’t work out that way. He doesn’t blame Veronica for what happens to the Followers outpost; he blames himself for not listening to House about what needed to be done.
He never really stops feeling guilty about destroying the Brotherhood, and he doesn’t expect her forgiveness. If anyone killed his family, even if it was for the greater good, he would hate them for the rest of their lives.
Lily: A good grandmother. Liam encourages her to keep taking her medicine at half-doses, because he can’t imagine anything worse than forgetting your own family. He mentally calls her năinai; fortunately, since he tries not to get drunk around her, he hasn’t yet said it out loud. He doesn’t like outright giving her orders, since she’s so much older than him.
Boone: Too reserved, grim, and focused on revenge for Liam to get along very well with him. Liam doesn’t really have any interest in traveling with an ex-NCR sniper, anyway, after the pressure the Republic put on Shi-Town to accept annexation following the ‘78 earthquake. He doesn’t do the quest One For My Baby, so it never becomes an issue.
Arcade: A genuinely good person whom Liam has a great deal of respect for. Liam likes anyone who genuinely tries to do good, and since he was with the Followers for three years before becoming a courier, traveling with Arcade feels pretty natural for him. Along with Veronica, Arcade is the human companion Liam spends the most time traveling with. (He’s very protective of ED-E, and does not appreciate suggestions about throwing him into a lake.)
Liam helps House keep absolute control of New Vegas, though, so that’s the end for them. He doesn’t ever talk about it with anyone, but he misses Arcade like hell post-game.
(And there’s at least a little awkwardness with the whole descendent-of-remanants-of-the-American-government vs descendent-of-Chinese-sumbariners thing.)
Cass: Liam doesn’t do the Crimson Caravan quests, so alas, he never travels with Cass. If he did, it would be an interesting dynamic. She’s blunt and never hesitates to speak her mind, whereas Liam is... not that.
Raul: An extremely cool ghoul. Liam loves listening to stories about the time before the War, though he doesn’t like them as much for a while after he gets back from Big MT. (And he feels terrible about what happened to Raul’s family.) It makes him slightly uncomfortable at first that Raul calls him ‘boss', seeing as Raul is over 200 years old, but he gets used to it. In Old School Ghoul, Liam convinces Raul to give up his gunslinging past and live a more peaceful life.