Yes, they are testing autonomous driving, but still legally is mandatory to have a real driver, and probably what they are doing now is training the AI and testing it against real driving conditions.

They specify their (very logical by the way) strategy here: https://www.moia.io/en/innovation

Because we are talking about a completely new technology with autonomous ridepooling, we cannot fully rely on existing empirical values. A special feature of the pilot project is therefore the early combination of technology, operation and customer experience. It allows us to test the operating procedures under real conditions in a safe and systematic setup. We will start in an initial test area in Hamburg: East of the Alster, it includes parts of Winterhude, Uhlenhorst and Hohenfelde and a route network of about 50 kilometres.

With complex road conditions and many traffic challenges, it offers ideal conditions for us. In addition to public transport, autonomous ridepooling can also relieve traffic congestion. To prepare for the deployment of the autonomous driving vehicles, it has already been started surveying (mapping) the service area. Intensive test drives with and without passengers as well as testing of the now increasingly automated tasks in the vehicle, which are currently performed by our drivers, will also follow up until 2025.

I've been in Hamburg and found these mini-bus. They are kind of passenger vans, but with specific interiors for easy operations for a bus-like usability. They are autonomous, but with a driver for test-legal-security reasons. Routes are on demand, not fixed. So it's like the Carriot service (long time ago, when web 2.0 was a thing) but now aiming for self driving. Interesting, among other things, is that it's a Volkswagen company, but a totally new brand and the busses-vans have no VW brand nor exterior design, they are full new designed vehicles.

It's a useful services no doubt, getting subsidies by the city looking for CO2/pollution reductions I guess, and about half the price of a regular taxi/cab ride for a single person (which used to be closer to one third when the service was new) but so far their vehicles aren't autonomous and the linked page doesn't claim them to be, nor are there other new milestones reported in this regard.