Buddies mode is a game mode that you can activate if you create another character, while having another character with companions met.
You gain an item called "Portrait of a Friend" on that new character, which upon activating in 12 in-game hours, allows you to pick an already met companion and start Buddies Mode with it.
On Buddies Mode, your character will have that companion always summoned, and you wont be able to dismiss it. Your character and companion will also grow stronger as the friendship level you have with that companion increases. Beside that doesn't stops you from having other companions following you, it will actually be harder for you to unlock more companions to follow you. Also, having other companions following you reduces the benefits from buddies mode, so the bonus status you and your companion would get reduces as more team members are added.
Once activated, buddies mode can't be disabled on that character, neither you can change who your buddy is, so choose wiselly who you'd pick as your buddy.