Amber let's you use Bunker login on Nostr web apps, on your phone or other devices. The relay connection is part of that. As for battery usage it is always <1% on my phone.

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And this relay.nsec.app handles this? Could I add my own relay here? Do you know of any good resource to learn more about how this works? :)

Yes, relay.nsec.app is the default for handling bunker/remote signing, but it can be configured to use a relay you choose. Not all relays accept the signing event types that are used for this signing method, though.

I go over both signing methods Amber uses in my full review of Amber at https://www.nostr-reviews.com.