@SeanStone Thank you for your workaround. Last week, we found a workaround for the problem as well. However, we cannot explain, why it was working at the end.
We Just had to delete the faulty layer in ArcGIS Pro, republish the service, add a new layer in ArcGIS Pro from scratch and setup all parameters according the old layer (naming, labeling, layer id, popup definition, definition query, symbolisation, etc.). After republishing the map service again, magically everything was ok. 🎉
At the end, we have an identical looking layer in ArcGIS Pro and on the server with the difference, that the popup is rendered.
We suspect a versioning and compatiblilty problem. The layer was migrated through several ArcGIS Pro versions and somehow ended in this currupt state.