No. Amazon Fire TV (Fire OS) runs Minecraft Bedrock Edition (v1.16+). Xbox 360 runs Minecraft: Xbox 360 Edition (Legacy Console Edition, last updated 2018). They are incompatible versions with no cross-play.
| Platform | Edition | Cross-Play |
|---|---|---|
| Fire TV / Fire Tablet | Bedrock (v1.21+) | Yes: Win10/11, PS4/5, Switch, iOS, Android, Xbox One/Series |
| Xbox 360 | Legacy Console (TU73 final) | No — isolated ecosystem |
| Xbox One / Series X|S | Bedrock | Yes — with Fire TV, Switch, PC, Mobile, PS |
Fire TV Note: Controller required (Bluetooth gamepad). Touch controls not supported for Minecraft on Fire TV.
Bottom Line: Xbox 360 Minecraft is a dead end for multiplayer. Friend needs Bedrock on any modern device. Fire TV is a Bedrock device—so they can play with Switch/PC/PS/Xbox One/Series, just not 360.