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.