Is there a way to enable emoji on TouchBar for every text field?
Does the Touch Bar support emoji for every field?

Yes.  The touch bar will support practically anything the developer wants to include.  The more appropriate question is, can you enable it?

Not unless the developer of the app allows you to.

The Touch Bar is an extension of the UI and is meant as another interface that the developer can utilize the enhance the experience.  See About The Touch Bar.

So, from the same document, you can see that emogis are available via the character picker but that character picker won't be available in the App unless the App developer enables it.