The documented gesture changed
Amazon’s input-label guide uses Menu on the highlighted input in its regular route and a Select hold for devices with the New Fire TV Experience. A different gesture in a current interface does not prove the remote’s Menu button is broken.
Confirm that you are on a Fire TV Smart TV input list and have highlighted the intended entry. A menu inside a streaming app is a different context.
Rename the verified source
Use the appropriate documented gesture, choose Change Device Name and enter a label the household understands. Before saving, verify that the entry corresponds to the correct physical cable and visible source.
Return to the list and select the new name. It should lead to the expected equipment; the label alone does not configure that device’s power, sound or player account.
Keep the scope to the television
A Fire TV Stick attached to another TV should not be assumed to own that TV’s input-label interface. Follow the television manufacturer’s route when the label belongs to another platform.
Record the interface version when asking for help. This prevents a correct older screenshot from being mistaken for an exact description of a changed current menu.
Questions answered
Why does my TV use a long Select press?
Amazon documents it for the New Fire TV Experience.
Does renaming configure the external player?
No. It changes the input label on the television.
Sources & further reading
Official references for the subjects discussed. Service terms and device software can change.