I'd hate a world where people claim they "own" certain UI layouts. It reminds me a little of when 37 signals were up in arms about huddlechat copying campfire, because both of them looked like chatrooms :/
Definitely if twitvid.io have a legitimate claim to the trademark, they might have a case, although afaik owning the .com counts for a fair amount. If it was me, I'd just change the name, knuckle down, and out maneuver the competition.