"Keep track of which number is managed by which operator" reduces to "store a small amount of information for each phone number".

In a traditional database setup, the "transactional method to update it" would be sending a request to a central server to update the database. The central server can be run by whatever regulatory body or industry association is responsible for number portability -- ideally, this would be the group with legal authority to overrule whatever the computer system says anyway. It wouldn't need to handle a large number of transactions per second because people don't change phone providers very often.

You could imagine doing it with a blockchain instead, but you could also imagine doing it using a Yahoo Groups email list -- that doesn't make it a necessary or good solution.