Downloads of rentals are optional. You can stream without downloading.
Any rental download will get invalidated after the rental period.
Downloads and partial downloads can be considered on iPhone, iPad, iPod touch, Mac, or PC. Depending on app settings, a download may start automatically.
Other devices wouldn’t support downloads of purchases or rentals.
(Re-)download TV shows and movies that you purchased - Apple Support
On which of these devices do you have the Apple TV app?
On which of these devices did you intend to download and watch this title?
Check your Apple Account device list to find where you’re signed in - Apple Support
Rent movies from the Apple TV app - Apple Support
Watch your rental on another device
You can watch your rental on any device that has the Apple TV app and is signed in with your Apple Account. After you download your rental on one device, you can stream the rental on another device simultaneously.
If you can't play your rental, see if the rental is downloaded or streaming on another device. To download your rental on a different device, remove the rental from your device, then download the rental on the device where you want to watch it.
One download max for rentals. One stream at a time max for rentals.