Revit needs to release the Type Catalog files after loading a family that use them.
That message makes the process of updating a type catalog tedious at best if you have to close Revit completely to release it so it can be edited, very inefficient. There was a brief period of time when Revit did release the file as it should, but that ended with 2018 if I recall correctly.
I tend to start a 2nd session while I'm working on the 1st, and keep tumbling to the next and starting another to move to next.
I read somewhere that it is intentional because of Edit Family and Export Family Types...to protect the existing file. The end result unfortunately is that it makes it quite inefficient to edit the types on the fly as a project evolves...the way most of them really do.
Does the type catalog push the updates back into Revit?
No a Type Catalog is passive, Revit only alters a family whose types are defined in a type catalog when the family is loaded again and the relevant types are selected again.
The purpose of a Type Catalog is to all a family to cover the entire range of options available but to only load the relevant ones into a given project. Otherwise we'd have to have all the possible sizes of every family loaded into a project all the time.