Started a new character yesterday and was just on my way from Azura's Shrine to Winterhold, when I saw a horse going crazy in the distance. Zoomed in with my bow and saw a battle was going on which, on getting closer, turned out to be a bandit attacking a travelling merchant. I was too late to help the merchant, but I could save the horse. (Not sure if it even needed me, lol.)

However, I though I might just as well give horses a chance this time. (svcks you cannot shoot arrows from its back ...) Since I didn't buy the horse, whenever I want to use it, it says "steal". Now I'm a little worried I might get into trouble, but so far, no one seemed to mind me taking it, no usual "stealing" reactions. Did the game realize the owner is dead or is it normal that NPCs don't care about stealing horses unless the owner sees it?

Is it possible to make the horse "officially" mine?

Not sure, but every time you mount the horse, you're committing a crime and potentially racking up a higher bounty.

You're probably better off walking.