Can an app installed from the Google Playstore install another app with different permissions?
It's not common but it's not unusual either. I use this MacroDroid automation app. It requires a helper app that can't be published on Play Store and therefore developer provides a link to download, with this explanation:

In order to toggle Wifi on devices running Android 10+ a new helper file is now required. This helper file is built against an older target SDK allowing it to still invoke the Wifi functionality. It cannot appear on the Play Store since they no longer allow apps built against older SDKs to be uploaded.

When a WiFi toggle action is triggered, the installed app (from Play Store) sends an intent to this helper app, thereby circumventing Android 10+ restrictions. So it's a win - win for all because the app functionality is not degraded.
So, if you trust the developer and are sure you need that additional app, you can go ahead. As an added precaution, download the apk and examine it's permissions before installing (Mixplorer does that).