Actually I think Microsoft would be happy for Sony to use it. You would have thought the same thing that Microsoft wouldn't want to help Sony by letting them use Azure servers for the streaming service, yet here they are.
I don't even think MS makes any money off DX. I'm nor sure how that all works.
Yes, MS would be happy to see Sony using it for PS but Sony first would always prefer their own stuff without the extra overhead that DirectX adds from the other versions, and also being Sony's stuff don't depend on somebody else to do stuff, and also saves them from having to share certain critical stuff in terms of security with an external company -and this case their direct competitor- like MS.
Also, the engine programmers who work on PS from Sony and the other publishers who make their own engines and the guys from the commercial game engines would always choose the most optimal, more optimized in terms of performance option. And I assume it would always be the option console specific one made by the console maker, in this case Sony.
Regarding MS making money from DirectX they don't make it from it directly. DirectX makes easier to develop games for Windows and Xbox, so they make DirectX to make sure there are more games published on Windows and Xbox. MS gets money from people buying Windows, Xbox, buying games for Xbox and their PC store and from GP. So the more games are there, the more money they have.