How do iOS apps from the same developer retain and access login information after being uninstalled without the use of iCloud?
The login information is securely store in the iOS keychain subsystem. Unlike macOS, keychain in iOS is not user accessible. However, apps can programmatically store and query for data in keychain.

Google developed apps look for pre-existing user information in the iOS keychain when installed. The login information is not removed from the device unless you choose to remove the account from your device from within one of the installed app.

iOS keychain provides a secure cryptographic disk store which is locked when the device is locked and unlocked when the device is unlocked. An app can access only its own keychain items, or those shared with a group to which the app belongs. Apps from the same developer, such as Google in this case, fall under same group (via common Bundle Identifier component) and this lets them share a secret via keychain.