Can the TARDIS be considered a companion? K-9 is certainly a companion so being biological isn't a necessary condition, and even if it were, there are several mentions of the TARDIS being grown rather than built. Captain Jack has a big glob of goo on his desk that he says will grow into a TARDIS in 50 years. The various doctors and their (One) TARDIS certainly have an interesting relationship. It's quite a bit different than a guy's relationship to his car which is what you might think it would be if the TARDIS was nothing more than a transportation device. It's more like a marriage, or like a boy and his dog, well, providing that the dog was way smart and way powerful, loyal and occasionally obeyed a command.
Before a TARDIS becomes fully functional, it must be primed with the biological imprint of a Time Lord, normally done by simply having a Time Lord operate the TARDIS for the first time. This imprint comes from the Rassilon Imprimatur, part of the biological makeup of Time Lords which gives them both a symbiotic link to their TARDIS and the ability to withstand the physical stresses of time travel (The Two Doctors, 1985) Sounds like a companion to me!
But the Tardis seems to go well beyond just taking orders from the doctor. In most of the episodes, when the doctor attempts to go somewhere, he randomly flips a few switches, hits the console a couple of times with a hammer (Percussive Navigation), and ends up going to a completely different place, but one where he's desperately needed. Most of the time (does the doctor ever say that?) it's something ordinary like the Daleks attacking the earth, or the Cybermen attacking the earth, or some other group of nasty aliens attacking the earth, but once in a while it's a bit different. "The Doctor's Daughter, 2008" is a good example. The TARDIS takes the doctor to the only planet where the doctor's daughter can and certainly will be born. Perhaps the TARDIS knows that the doctor's daughter will be needed in a future that only the TARDIS can see. Another one of these is "Tooth and Claw, 2006." There the TARDIS misses the target year by -100 and ends up in Queen Elizabeth's time where, as a tiny sidebar to the story, ends up causing "The Torchwood Institute" to be created, which comes in handy in later stories. Yes, I know this story was just the BBC writers preparing for the Torchwood Dr Who spinoff, so that would mean either that the BBC owns the TARDIS or the TARDIS owns the BBC. Let's see, which one of these two is smarter?