
Yes, You Can Adopt a Dog in Oblivion Remastered. No it Can't Sleep on the Couch.
Emily

There’s a reason websites like ‘Does the Dog Die?’ and accounts like ‘Can you Pet the Dog?’ exist - we bloody love our pets. We love our pets, and we all love animals in video games. Not only dogs but cats, horses and even (in the case of Shadow of the Tomb Raider) llamas. The ability to interact with animals in video games adds wholesome interactions into often violent gaming landscapes. It’s also really cute.
In Elder Scrolls IV: Oblivion Remastered (unlike Skyrim), you sadly can’t adopt a dog without mods. However, you can get a canine companion. Sort of..

When venturing in the Shivering Isles DLC, you’ll eventually end up at the town of Crucible. There you can find a scaredy cat called Ushnar who’s, well, afraid of cats. His life is a misery thanks to a mischievous Khajiit who spends his days following Ushnar and his dog around the streets of Crucible, simply being a menace.
By inserting yourself into this childish game, you can gain Ushnar’s favour, and the reward is interesting.
First, you need to be on good terms with the Ushnar - enough that he’s happy to share his woes with you, and then you need to decide how to get rid of Bhisha. That’s completely your decision. There isn’t a penalty for killing him (unless caught, of course), but you could equally just convince him to move elsewhere. Something he’s actually quite happy about. So that's nice.
When you speak to Ushnar, he has quite the strange response...

Not long after completing the quest, your new best friend will arrive! And minus the skin and eyeballs, he acts just like any other pooch. Loyally following you around the Shivering Isles and Cyrodiil. Be aware, he doesn’t fast travel with you; he’s just super speedy, so if you end up somewhere without him, don’t fret, he’ll find you.
So, now the sad part.
Unfortunately, you can’t pet the dog. You actually can’t interact with him at all, telling him to stay put or to meet you at home. This means that he does, in fact, always follow you. Even in dangerous situations. And the even sadder part? He has a very low HP, so he is easily killed. And he won’t respawn.
But having a dog for a day is better than having no dog. Right…?
Sometimes it pays to play on PC over console, as Nexus Mods has the key to keeping your poor pooch alive as well as making him a bit more fun to have around. For the rest of us, though, we have to just appreciate the time we had together and hope the skinned hound made it to Sovngarde. RIP, little buddy.