It seems that I've found a solution in my case. You must go to the options of the Enpass application (desktop) and open the Browser options. Then UNCHECK the switch "Authorize browsers With Verified Code Signatures Only" and CHECK the switch "Ask for Browser Authorization Every Time". Then go to your browser extenstion and try to open it. There should be no error now but question about authorization code. You input the code that is displayed in the browser and the extensions starts to work again. Then you can switch on the first switch and done (at least in my case). I can now again autofill in browser with the hotkey.
UPDATE: Switching on the "Authorize browsers With Verified Code Signatures Only" after the authorization causes that browser doesn't work again and displays the 303 error. So I consider this as a bug. Please fix it @Yogesh Kumar