I think Google authenticator implements the standard OTP which lots of apps (including keepass) should support.
Microsoft uses their own propietary crap
You can try to add the standard OTP even for Microsoft crap. If it asks you to register for mfa and opens the screen that says something about downloading the Microsoft authenticator app there is a small link at the bottom, letting you use another app. Then you get a qr code that you can scan with any other auth app.
I use a basic OTP password instead of Microsoft's ironically less secure (see SMS as 2FA) with my work MS account. Perhaps your org disabled it but it is definitely something a Microsoft account can do.
Proper Microsoft authenticator setup is more secure than OTP because it's pushed based and doesn't allow users to copy paste their OTP codes into phishing sites. Google also prefer push based MFA for this reason.
Push based, sure. Allowing SMS, I still hold, undermines all of this.
They "secure" this behind password which you entered to trigger the SMS push in the first place.
Offering an "out" to a more secure flow means your secure flow may as well not exist.
Additionally, phishing a pushed OTP is not really much harder since you can trigger the push and then just have the user finish off the flow for you, provided they don't read the IP or whatever you display them (they won't, they think they're signing in), effectively the same as a TOTP.