No. Even allowing for the uncertainty about who counts as a companion, there have been several undoubted companions who were not human. K9 ("metal dog"), introduced in "The Invisible Enemy", is the most obvious but there have also been:
- Susan (Time Lord/Lady - or, at least, Gallifreyan), introduced in the first episode;
- Romana (Time Lord), introduced in "The Ribos Operation";
- Adric (Alzarian), introduced in "Full Circle";
- Nyssa (Trakenite), introduced in "The Keeper of Traken";
- Kamelion (android - debatable companion status), introduced in "The King's Demons"; &
- Vislor Turlough (Trion), introduced in "Mawdryn Undead".
And that just takes into account TV companions: in the comics, novels, and audios the Doctor has travelled with a number of other characters from different races and species:
- John and Gillan, presumably from Gallifrey, introduced in "The Klepton Parasites" (TV Comic comic story), but revealed to be dreams in "The Land of Happy Endings" (Doctor Who Magazine comic story).
- Frobisher the shapechanging Whifferdill, introduced in "The Shape Shifter" (Doctor Who Magazine comic story);
- Compassion the Type 102 TARDIS, introduced in "Interference - Book One" (BBC Eighth Doctor Adventures novel #25);
- C'rizz the Eutermesan from another universe, introduced in "The Creed of the Kromon" (Big Finish audio story)
- Kevin, a sentient robot dinosaur, introduced in "When Worlds Collide" (Doctor Who Adventures comic story); and
- Decky Flamboon, a Brancheerian from Sirius, introduced in "Meteorite Meeting" (Doctor Who Adventures comic story).
And of course "The Doctor's Wife" confirmed that the TARDIS is a sentient creature who has been the Doctor's faithful companion for centuries, introduced in either "An Unearthly Child" or "The Pilot Episode", depending on your viewpoint.