One Fire Stick will work on multiple Televisions but it will only work on one TV at a time.
If you have a few different TV's in different rooms but you only watch one TV at a time then yes you can move that one Fire Stick From TV to TV and all your settings will still be intact on the device.
If you want to stream movies on multiple TV's with the Fire Stick at the same time then you'll need a Fire Stick for each TV.