$TERM is special – it is not sent as an environment variable (which wouldn't make sense for non-Unix systems anyway), but rather as a special "terminal type" field when establishing a pseudo-terminal channel, per RFC 4254 § 6.2.
6.2. Requesting a Pseudo-Terminal
A pseudo-terminal can be allocated for the session by sending the following message.
uint32 recipient channel
string TERM environment variable value (e.g., vt100)
uint32 terminal width, characters (e.g., 80)
uint32 terminal height, rows (e.g., 24)
uint32 terminal width, pixels (e.g., 640)
uint32 terminal height, pixels (e.g., 480)
string encoded terminal modes