What ancient version of Notepad++ are you using? View > Current File in > Edge has existed for years now – it was added in v7.6.5 in September 2019. It was only versions before that in which you used the Run > Launch In XXX entries.

And if you still need to add some other browser, then yes, you can add any external application you want, as long as you know the command line; it’s really easy: Run > Run…, type the path of the application to run (and maybe use the variables described in the Config Files > User Defined Commands section of the user manual, like using c:\path\to\otherbrowser.exe "$(FULL_CURRENT_PATH)" to run the currently-active file in otherbrowser – then Save that user-defined command.

(Those Run > Launch In entries were removed a little earlier than the alternative was developed, in v7.6.3. This was part of the EU-FOSSA effort to avoid security compromise vectors from being distributed by default.)