As a result of a Door series one player not having enough players to run his game last night, he created a cameo character for episode 3.02. Rather than someone from the adventure's setting era, he brought in another Time Lord. After the rather convoluted "he was in a different time warp when the lock closed and technically isn't from this universe" explanation for Kai, this one was handwaved. But as SteveD notes, "Also, this is Doctor Who: canon is the thing you leave wreckage of behind you as each new episode arrives."
He was designed to be a contrast to the eccentric enthusiastic occasionally moody Kai, a professional from the Gallifrey Battalions, purposely colourless. Having his own TARDIS means he can drop in and out of the game on a session-by-session basis.
His TARDIS was never shown on-screen - or it was, but its chameleon circuit works properly so it could look like anything. But while not an issue this session, it did raise the question of what to do in a game with multiple time machines available. It makes "get to the TARDIS!" solutions and escapes simpler, for one thing. "Splitting the party" could become more problematic if they're in different parts of the universe and different eras.
Of course, these days "universal roaming" means that people aeons apart can phone each other. At least, Rose and Martha could phone the days they were "supposed" to be at. As of series five the TARDIS phone can field calls from across time and space, so maybe that issue is confined to mobiles.
By comparison, having another Time Lord around is no big deal (apart from the continuity trauma, and what's Doctor Who without a little continuity trauma?) as he's far from Doctor-level, just a bloke with a temporal scanning gauntlet that may also contain a personal forcefield.
But yeah... a spare TARDIS floating about could be tricky.