Princess is a character from Super Robo Mecha Force Five!, a fictional anime that the turtles watch in the 2012 TV series. She is the princess of an entire planet and pilots Super Robo Mecha's left leg. Dr. Blip has a crush on her, and the two are supposed to represent April and Donatello.
Not much of her character is explored, but it's implied she may be vain and is very sensitive. This vanity was shown when Dr. Blip tried to ask her out to dinner, but she harshly rebuffed him for ‘being a weakling,’ although she later kissed him after he saved her life. In Target: April O'Neil, she left the team because the team forget her birthday. It's unknown what else has caused her to leave, but it's likely her vanity and sensitivity are part of her constant leavings. In the same episode, Leonardo mentions that the princess has left the team more than "twenty times" and always returns, to which Michelangelo has indicated that the show "paradoxes" Donnie's life (although he meant "parallels").
However, she also seems to be kind-hearted and peppy, as she constantly cheered on her team. She is apparently the cheerleader of the group.