Here are two theories, but they both eventually end up in a respawn chamber (which is canon) or some other apparatus that handles regeneration.
1st let's establish that regeneration, cloning, and consciousness / brain transfer are canon in the Titanfall universe. We also establish that the biodata cards you use to respawn teammates, and the same cards Crypto collects, are canon (check launch trailer).
Now we can discuss the two theories about the apex games / legends while in a match.
Theory 1: We're all clones of the prime legend, and after an apex match, all the biodata cards are collected, and consolidated in the prime legend for use as a source for the next match.
Theory 2, which holds more weight imo: The prime legends are the ones themselves in the match. We as players just play multiple copies of the legends for gameplay reasons. This theory holds more weight since we always see only the prime legends being dropped from the dropship during a match. They're then respawned via their data cards at the discretion of the syndicate.
Some legends do die, and leave in body bags (as per Gibby). If the syndicate doesn't feel like a legend will provide an ROI, they just don't ress them. Dead.
This ties into Gibby's lore where he joins to save his friends (bec they might not get ress'd), and also Apex lore where it's confirmed legends actually die, and explains how you can build a career in a blood sport where death is a thing (Mirage apparently being one of the most famous legends). The respwan chambers (again, canon) tidy up the whole 'death' problem of the Apex games, and conveniently also fit into Titanfall lore by providing a label for whatever apparatus is used for regeneration.
Legend contracts are in place to assure (in a way), that a legend is at least guaranteed 1 regeneration should they die in a match. Though I can see the syndicate just saying "nah. they're dead. who's gonna sue us?" Legends that help ratings and turn a profit are ress'd, and those that don't, aren't.
A special case is Rev, as he was put in the games for containment reasons. No matter how many times he dies, a new simulacrum seems to show up. So hammond may be hoping that by killing him over and over and over again, he eventually runs out of copies. They don't know he has his own personal manufacturing facility underground. They may be incapable of tracking / tracing the teleportation source of his new copies after he dies in the games. This ties into his latest lore bit where he wonders how he woke up underwater.
Oscar mike ladies.