Following the docs https://docs.jabref.org/collect/jabref-browser-extension for Edge I had the following Errors/Warnings:

Extension options tab:
Connection Status: Access to the specified native messaging host is forbidden

Console error (When using the extension on ArXiv)
TypeError: Cannot read property 'items' of null at [...] logError @ zotero.js:335

Workaround solution

1. CMD: REG ADD "HKCU\Software\Microsoft\Edge\NativeMessagingHosts\org.jabref.jabref" /ve /t REG_SZ /d "C:\Program Files\JabRef\jabref-chrome.json" /f. Note: \Microsoft\Edge\ (not described in the docs);

2. Open the Extension Options tab and copy the string after "?options=". It should be pgkajmkfgbehiomipedjhoddkejohfna. (different from the Chrome one);

3. This string should replace the one in the jabref-chrome.json file, that becomes:

{
  "name": "org.jabref.jabref",
  "description": "JabRef",
  "path": "JabRefHost.bat",
  "type": "stdio",
  "allowed_origins": [
    "chrome-extension://pgkajmkfgbehiomipedjhoddkejohfna/"
  ]
}

So, I would add a jabref-edge.json file with the correct extension ID and update the docs.