If the player already has a companion of a specific type, they need to be dismissed through dialogue. The player can have one humanoid and one non-humanoid companion (Rex or ED-E) in their party at a time, with both providing their unique companion perks (unless using exploits).
Companions are controlled via the Companion wheel. From left to right: Rex, Rose of Sharon Cassidy, Raul (Vaquero outfit), Lily, Boone (assault armor), ED-E, Veronica, and Arcade.
All eight permanent companions in Fallout: New Vegas.