In BF3, can you steal a vehicle that has enemy players in it?

Yes you can, although it's almost certainly a bug. (The video was meant to disprove an earlier version of the same bug, where anyone who stole a vehicle with an enemy still in the co-pilot seat would be flagged a member of the other team, and therefore invincible unless friendly fire was on.)

This just happened to me.

I was the gunner in a tank on Kharg Island, then out of nowhere the driver name became RED and I was in an enemy vehicle. It would not let me stay inside and I was automatically ejected after 5 seconds

No, its not possible. Unless there is a bug or a glitch i have missed.

It is possible. I have done in BF3 in conquest on the Gulf of Oman map on RU team, at US aircraft carrier in their venom. Basically I parachuted to their base then got in the venom and 3 seconds later there was a white name as a gunner.

I check the score board and notice the white named person wasn't on my team and that it was an enemy. After driving the stolen venom back to main land I get destroyed by the viper. I still don't know how it happened and I know it was a glitch and not a bug or mod.