Can the same player win an initiative or monarch trigger multiple times in the same turn?

Both the initiative and the monarch are continuous designations — only one player holds each at a time. Each has two relevant triggered abilities: one that triggers when you become the monarch/take the initiative, and one that triggers at the beginning of your upkeep while you hold that status. The upkeep trigger can only fire once per turn naturally, but the 'becoming' trigger can fire multiple times in the same turn if the designation changes hands and then returns to you.

For example, if you take the initiative, then an opponent takes it from you (via a dungeon-exploring effect or combat damage), and then you take it back again in the same turn, your 'you took the initiative' triggered ability fires twice in that turn — once for each time you became the initiative holder. Each time is a separate event that independently satisfies the trigger condition (CR 603.2).

The upkeep trigger ("at the beginning of your upkeep, if you have the initiative / are the monarch") follows normal 'intervening if' rules (CR 603.4). It only triggers at the specific point in your upkeep step, so it can only trigger once per turn under normal circumstances — it won't retroactively trigger again just because you regained the status mid-turn.

Concrete example: It's your turn and you currently hold the monarch. During your main phase, an opponent's instant effect makes them the monarch, then you immediately play a spell that returns the monarch to you. Your 'you became the monarch' trigger fires again. You've now accumulated two 'you became the monarch' triggers this turn (plus one upkeep trigger from earlier), letting you draw cards for each.

There is no rule that prevents a triggered ability from triggering more than once per turn unless the ability itself contains a restriction like 'once each turn' (CR 603.2, 603.4). Neither the initiative nor the monarch abilities contain such a restriction on the 'you took/became' trigger.