How to I submit a run so that the second player has his name colored?

I don't think you can,sorry.

Like in 2 player runs, there are 2 players listed. My name is colored, but my brother's is not. Look at level 32 2 player any% 1st place to see what I'm talking about.

Maybe there is an option/setting for it. IDK because I don't play 2 player Any

Your partner will need a speedrun.com account if you want them to have a custom color name.

He does. I put in his exact username.