Q: App asks for Apple-ID password; Is that legitimate?

you'd need to ask the developers, but more importantly you'd need to trust them

IMHO trust is always earned before it's freely given.

I have never used a 3rd party app that asked for my AppleID password, and I never would.  I assume this is because "App for WhatsApp One" provides the function to FaceTime call directly from within the App?  Even so, they should not need to cache or store your AppleID password as they could simply have FaceTime prompt for it when needed.

To my way of thinking, that's like a credit checking app asking for your Bank login password.  FaceTime already has my AppleID password cached itself, from when I first signed in to the service on each app, so why would any 3rd party app need it just to launch an Apple App or service?

P.S. Apple's own security suggestions run counter to ever sharing your AppleID password with any 3rd party (app, person or web site) for any reason - Security and your Apple ID - Apple Support

That I cannot say as I do not know the details of Apple's App vetting process.  I do know there have been 3rd party apps for eMail, and if those support iCloud IMAP mail, then you have to provide your iCloud password to the app for it to access your mail.  So it may not be inherently forbidden for any App to request it.  I just know I would not use any such app though (and I use both 2-step verification and 2-factor authentication with my AppleID, but I still see no reason to ever give those credentials to any third party).  I would either simply live without whatever features the app offered, or find an alternative.