Havenite teams are named after heraldic creatures. More fearsome creatures are considered to be more prestigious. It's actually tradition for graduating teams to bestow their team name to a freshman class as a sort of mentorship relationship. This is supposed to be based on the performance of the freshman team, but as most things are in Mistral, it's more commonly based on relationships between noble families and familial standing.
In theory, a team could be so good that they should be granted the Phoenix as their team banner, but unless they're from important families or have strong backers, in practicality the best a team of extremely talented no-name hunters in training could get would be Eagle or something.
Telamon Kolouris is of noble descent and controls important harbor holdings in Mistral. For this reason he has been considered an extremely important player in the last few years. His brother, and Pyrrha's father, Peleus, is captain of the Myrmidons, a security contractor that acts as Kolouris's private security.
Pyrrha is on the outs with her uncle. Her father is ever supportive of her, but by refusing to attend Haven, she's screwed up the team arrangements that her uncle had planned. He's not abandoned her, she's way too skilled and valuable for that, but you may see him in later segments attempting to pressure Pyrrha into returning back into the fold.
Pyrrha's opinion on Mistral's familial politics is that they're stupid. Family doesn't mean you're great. Your own prowess and skill mean you're great.
"Imagine a king who fights his own battles. Wouldn't that be a sight?"
EDIT: oh you mean the actual name thing
nah that was stupid i threw it out