HKX actually did have a contract with DB Regio for mutual acceptance of tickets for a while, but they withdrew from it, I believe because they found their services were becoming overcrowded by DB customers.
It doesn't really make sense to talk about the "DB ticketing system". DB Regio and DB Fernvekehr(long distance) have completely different fare structures, and in almost all regional trains tickets from the local tariff associations will be valid, so for any given DB Regio train you might have bought your ticket from any one of 3 different people, according to 3 different fare structures, 2 of which can be changed at any time by Deutsche Bahn according to their commercial decisions.
The area where the "being part of DB" concept makes more sense is that the Deutsch Bahn journey planner is the de-facto national rail journey planner/ticket vendor for rail services in Germany, but will not sell tickets for these services, only display the timetable. However, FlixBus now has a near monopoly on long distance bus services, so a vast number of people are going every day to the FlixBus website, where they will be presented with these services alongside all the bus results.