English Russia website said "Moscow ecologists think that dogs started acquiring this habits in 1990s, when the Soviet union collapsed and Moscow fell into the hands of new class of Russian capitalists." The homes that they used to shelter in were removed to the outskirts "so the dogs had to move together with their houses, so they had to learn how to travel Moscow subway - first to get to the centre in the morning then back home in the evening, just as us people."
They certainly seem at home on the subway and if you look at the video at the end of the post, some of them seem, well..... a little too much at home.
Fortunately I don't think there are enough stray dogs around London to see them using the Tube by themselves. I could imagine that cats would be sneaky enough to give it a try, but as they're not as social as dogs, I think we can rest assured that our feline friends won't be cat napping on the Tube with us.