Is there a way to get another dog?

There appear to be only two companion dogs in the game:

• Meeko - you can find this dog in Meeko's Shack
• Vigilance - who is the dog you've described.

Commenters point out that there's a stray dog you can find, and that during a Daedric quest you will temporarily gain an "essential" (and therefore unkillable) dog follower, which you can keep as long as you don't finish the related quest.

You can only have one at a time, and when they're gone, that's it for dogs.

Ya, like dags? Dags?

The answer is yes, you can get more dogs:

• Meeko - One only. Lame
• Vigilance - One only. Lame
• Barbas - One only. Super awesome, invincible dog (but also annoying)
• Stray dogs - These can be found in the wild multiple times but they're pretty lame.