I've the mobile version of metamask installed on my phone and I've never seen anything like this. Also, I've gone through some replies but your explanation makes a lot of sense. The op didn't download the app from playstore because if he does he wouldn't have gotten such a message prompt asking for access. My guess is, he used a phone browser such as chrome to download metamask from the official website, and when he was trying to install it, he got the message because the phone recognizes playstore as the only verified source for installing apps, any other sources like phone browsers needs to be granted access.
Playstore doesn’t guarantee an app is genuine, it only means an app is registered following due procedures but, there are a lot of mirror apps on playstore too though, there is a likeliness that, you could get suggestions on what is genuine based on the most downloads on an app of interest.
Also, apps that raise concerns are only later attended to as per deactivation from playstore.
I’ve been off metamask for a while now but, I don’t think it did ask for such permission for modification or any permission at all while I used it and besides, it’s supposed to be a wallet and not some all controlling app. Wallets don’t seek permissions for modifications, not even an upgrade.
@OP, you do have to be skeptical and it’s good your not just ‘Yes and Okay’ about all your presented with.