I think most replies are missing the main point OP is asking about.

The answer is yes, it matters. I started a run as Gale to learn the basics of the UI prior to starting playing. You learn an important (secret, to begin with) detail about Gale (a certain condition) as you as your exit the nautiloid.

On the other hand, playing as a custom character, he first told me about that 15 hours in, and even then only in very vague terms and refused to give more details when pressed.

So when you start as an origin characters, you really "are" that character. When they're just a companion, they'll keep secrets from you.

In some cases it really matters who is initiating the conversation and I think that way you do get some of the information of their origin story. I personally choose who I think is suited best to lead in the situation, sometimes even when I lead with a companion, the game still initiates with my custom character, I then have to unlink the group and try again.

I have no idea how the game decides or behaves in coop.

I'd assume for the best origin story experience and ending, you should go for the origin character as your "main" character. But I haven't finished the game myself yet.

Who you have actively selected and control will determine what stats are used during conversations/cutscenes, IE if you need to do a charisma or intimidation or insight check it uses that character's stats. However certain cutscenes will use your main character (the one you selected when you started the game) no matter what.

Also passive checks throughout the world will use all characters but they need to be near the spot to use them so sometimes if your character you're controlling walks past a hidden cache somewhere and fails a perception check you can move your other characters into that spot and they'll all try. You can even go to camp and grab someone else for more chances if you really have some blind characters in your party.