Possible doesn't mean easily doable, unfortunately. Technically speaking, I don't know how hard it would be for the fediverse. I get the sense that overall it's been a mismanaged aspect of the ecosystem for a long time.

It touches on a broader issue of to what extent the software ecosystem enables users to exist on the fediverse at large as a single user or through a single interface. At the moment, it's basically not really a thing. Arguably, if the fediverse wants to make any claim to being an actual "federated universe" rather than just separate FOSS decentralised platforms (there is a big difference IMO) ... then it should definitely be a thing.

In relation to DMs, then, in a "true fediverse" the answer would be simply something like integrating matrix into your interface such that you and I could easily start a space on matrix and start chatting there if we wanted to.

I've come to the conclusion that for this to happen it needs to happen at the UI/client/app level. In fact, I wouldn't be surprised if that happens in not too long a time. An app that understand and works well with all of the major platforms and gives you a single and well designed interface for working with all of them from a single space. This way the platform developers can focus on their specific funcionality and backend while the app/client developers can focus on the UI and the challenge of bringing things together. I see it as similar to the way we all have email apps that easily bring together multiple email clients.