Is it possible to install already installed apps with one click? In the Apple App Store, if you have downloaded an item in the past, you can click the INSTALL button to download it in one click. However, in the Google Play Store, you have to go into the app, press the install button, accept the permissions, and then it's downloaded. It's a bit tedious if you're batch re-installing after you did a factory reset. Is there a function similar to that button in the App Store, but for the Google Play Store?
This doesn't specifically exist, however there is the Android Backup System, which allows you to reinstall all the apps from the Google server's on the inital setup of the device.

This will queue all the apps for install from Google Play. It comes on all Android 2.2+ devices, however it is very temperamental.

Other than that, you could use a dedicated backup app such as Titanium Backup Pro to regularly backup your apps, allowing you to restore them in one click should you need to.

Google Play doesn't offer what you stated. iOS offer's this because they don't have permissions to show you ;)