Gmail passwords don’t show up in Keychain Access. Are they saved somewhere else? I usually use Keychain access in Apple devices to locate forgotten email passwords for Outlook for Mac.
The Microsoft ones and others appear in Keychain Access.
However, the gmail passwords don’t show up.  Are they saved somewhere else?
I use this trick to recover forgotten passwords in customer’s Macs too.
Any help much welcome
I've tested it out and it seems that Microsoft Outlook for Mac sets up a sort of automatic authentication system. I didn't even have to put my password in it just logged me in.  I assume it got my Gmail password from the Keychain as I had it setup with Apple Mail. I assumed wrong. I also noticed the password field is not filled in on the account settings for Gmail in Outlook for Mac.
It seems Google Supports OAuth2 and Microsoft implemented it.
https://developers.google.com/gmail/api/auth/about-auth
https://developers.google.com/identity/protocols/oauth2
High level overview of OAuth2 and how it works:
https://youtu.be/CPbvxxslDTU
OAuth2 allows for credential-less authentication. When you setup your Gmail account in Outlook for Mac (likely PC as well) it asks you to allow Microsoft to access your Gmail account. Then you never even have to provide the password.  Google is doing to check to ensure you are who you say you are. If it knows this already it just lets Outlook login. If not it will have you authenticate but credentials go to Google not Microsoft. Then next time it connects it doesn't even send the ID/Password.
The Gmail password is not saved in the Mac Keychain as a result. If something breaks the chain of trust I am sure Google will prompt for your credentials, etc.  I didn't even have to use my Multi-Factor Authentication either.
Try adding another Gmail account or changing to a new user profile in Outlook for Mac.  Go to /Applications and right-click Microsoft Outlook then Show Package Contents. Go to Contents/SharedSupport and double-click the Outlook Profile Manager. Create a new profile and make it the default. Restart Outlook.  Add the Gmail account. In order to rectify any authentication problems you work with Google this way. i.e. Forgot Password, etc.
Creating a new profile will not affect the old profile. The data for a profile is kept here:
/Users//Library/Group Containers/UBF8T346G9.Office/Outlook/Outlook 15 Profiles/
If you drill down that path your old email should be in there and you could import it back into the new profile.