How do you claim beta eggs at launch?

The account is the part that matters. Whatever device or character you end up on, the claim is tied to the account you used during the closed beta, and logging in with a fresh account will not find the egg. The egg is not handed to you in game either. You go to the official event page, sign in there, and claim it.

What is inside was decided during the beta, not at launch. The Bonded Promise event let each participant pick one Aniimo to carry forward, so the egg holds that single choice. There is no second pick at claim time.

Why the egg may not open straight away

A sealed Aniimo can only be reclaimed once you have unlocked that Aniimo's corresponding form in the Aniilog in the live game. The live account has to catch up to the beta choice first. If you sealed something you had not touched since, expect the egg to sit unopened while you work that Aniilog entry.

What could not be sealed

Prismana and other special Aniimo cannot be sealed inside Promise Eggs as carryover rewards. In the beta both starter creatures were classed as special Aniimo, which is why neither could be selected for the Bond Pact sealing event. If you were hoping a starter or a Prismana was waiting in your egg, it is not.

Not settled: The full roster of creatures available at launch has not been confirmed, and reclaiming still depends on unlocking the matching Aniilog form in the live game.