TL;dr:
My discovery process is kinda listed below.
https://www.slashgear.com/google-android-app-beta-makes-it-easier-to-share-search-results-20581224

MarkMonitor.

Corporate Domain Management

Your brand portfolio is exceptional. Shouldn’t your domain management service be the same?

Looks like they are a domain squatter, buying up domains and selling them at ridiculous prices.
They have a page showing some domains they have for sale https://www.markmonitor.com/domains-for-sale/top-level-domains/
But I don't see search.app listed. Doesn't mean they don't own it tho, or perhaps they managed the acquisition of it.
It's strange, because it seems like Google Domains is the registrant:
Registrant Organization: Google LLC.
Maybe MarkMonitor owned it and leased it to Google?

search.app.goo.gl probably also points to the same firebase app: https://websecblog.com/vulns/bypassing-firebase-authorization-to-create-custom-goo-gl-subdomains/

Both the Google subdomain and the TLD point to firebase hosting.

Firebase is essentially free hosting (and some Backend as a Service things).
I can't find any details on who is behind it tho, and I don't think there is any way to publicly find those details.
I'm guessing it's some sort of link obfuscation or shortener service.

It might be that it is an official Google service for their apps, which is why they are the registrant.

Ah, found something:

https://www.slashgear.com/google-android-app-beta-makes-it-easier-to-share-search-results-20581224