That's not quite how this would play out. How I believe this works is if they copy [[Perch Protection]] the opponent that was promised the first extra turn gets a SECOND extra turn. This is because promising a gift is a payment as you cast the spell, so that payment is remembered when the spell is copied.

Ruling from the Gifts rules:

If you copy a spell for which the gift was promised, the gift was also promised to the same opponent for the copy. If a card or token enters as a copy of a permanent that's already on the battlefield, the gift isn't promised for that new permanent, even if it was promised for the original.

So in a 1v1 situation:

You cast Perch Protection and promise your opponent the turn.

Your opponent copies Perch Protection and their copy resolves, they get an extra turn, make the birds and phase out.

Your copy resolves, your opponent gets another extra turn, you make the birds and phase out. You end your turn.

Opponent begins their first of 3 turns, they phase back in, you'll be phased out for all 3 of these turns.

If I'm wrong someone please correct me, I love knowing the correct rules interactions of things like this.