modify the action:
If you create an action to save for web with PNG file, Illustrator don't override the save command and nche the action take the same name of the file in the action.

try to open 5 or 6 files and create the action:
New action > save for web > save (and stop action)
Now use it on the other opened files...

If you have more and more files like me and your client ask you to export the file only in png it's impossible to use this action!

The only way is to save files one by one!
This is crazy!