Why pairing is per machine, not per account
Pairing only ever grants control of the one specific machine it was set up on, never your account as a whole across every device you own.
If you use tCode on more than one machine, a laptop and a desktop, for instance, each one needs to be paired separately, with its own distinct code generated from its own Remote Control chat.
This per-machine scoping is deliberate: it means losing or compromising one paired machine does not automatically extend remote control to every other machine you happen to also run tCode on.
Each paired machine also gets its own Remote Control chat in the web app, rather than sharing a single combined feed across every device; this keeps a laptop's remote-triggered turns and a desktop's remote-triggered turns from appearing interleaved in one place where they would be easy to confuse.
There is no limit described here on how many machines can be paired to a single account at once; each simply needs its own code and its own explicit /pair command run on it.