Q:

Could the EU introduce an app-crossing messenger standard and make its use obligatory for every messenger?

Let's assume the EU does the following things:

It develops a messaging standard for texts, speech messages, documents, video calls, end-to-end encryption and more and makes this open source (like the Email-standard for emails). This standard allows communication between apps by different developers. It could be an extension of the matrix communcation standard or Rich Communication Services (RCS) protocol.
It makes it obligatory by law for every messenger (e.g. WhatsApp, Skype, iMessage ...) to use this standard. 

This means that every use can use the messenger client of their
choice to communicate with every other user independent from the
app/client they are using. For example I could write an iMessage in a group chat with other users using WhatsApp and Skype respectively. Clearly that would be a huge consumer benefit increasing consumer welfare and it would foster competition between app developers. 
Would the EU be allowed to do so and make the use of such a standard obligatory instead of voluntary?

A:

Three questions here:

Could any state require such standards?
The state might have signed international investment agreements to protect foreign investors from sudden non-tariff trade barriers. The owners of the messenger services could demand compensation. Or there are bilateral or multilateral treaties on the trade in services.
Another problem might be constitutional issues regarding the freedom of speech. Does the state have the right to stop their citizens from using foreign-hosted, non-compliant services?
Could any state enforce such standards?
In the first bullet point, I asked if the citizens have the right to use foreign messengers. Even if the state thinks they do not, could it stop them from doing so? China tries.
A less intrusive way to go about this would be to go after the money. If the message service makes advertising revenue in a state, there will be an office where court orders can be served, or accounts that can be seized. The corporation could try to organize to make this more difficult, of course.
Has the EU been given authority in this area by the member states?
Yes. They were allowed to pass the GDPR, which includes provisions for data portability. What you propose is just a more real-time use of the same principle.

