Personally speaking, I wouldn't own a dog if I didn't have a car. That's not to say it's impossible, but it would make (my) life a lot harder.
With a car, I have the ability to take him to the vets if needed. It means I don't need to worry about taking an injured or vomiting dog in a taxi or on a bus (or try and get them there on foot).
I can drive him to all sorts of places - the beach, the woods, the nearest town, the pet shop, the groomers. It's quick and easy and means things like getting his nails done only takes 20 minutes as opposed to 90 if I had to walk there/back.
It means we can take him on holiday or on day trips without worrying about him getting muddy, or rolling in fox poo, or getting injured and not being able to walk home. We can go to the pet shop, or the nearest town to work on socialisation etc too.
BUT we do live rurally and I'd probably feel differently if I lived in a larger town or city, or at least somewhere with decent public transport.