First, Mods, if this breaks any rules, please lock this puppy down!
Second, I know it is illegal to share copywriten songs. But what about music videos? Yes, I know they are copywriten, but they are not available for sale that I know of.
Third, I apologize if I am confusing at all - I’m on some pretty strong cold meds right now!
I think we can thank Snap! for incorrectly saying “copywritten lyrics so they can’t be stolen…”
The tense of the word “copyright” that you should use is “copyrighted,” not “copywritten.” It’s not a copywrite, it’s a protection of RIGHTS for COPYing. Make sense?
Anyway… to give an answer to your question, they are also copyrighted. Anything you see on TV may be recorded for PERSONAL use, but you’re not supposed to make copies and hand them out to all your internet pals. Some shows/videos end up being put on video for sale later. So, in general, everything is copyrighted from the get-go. So, if you don’t feel comfortable downloading music, then feel the same about videos. If, however, you feel fine downloading music, feel free to grab up some videos. 
For sale or not, a copyright is a copyright. I know of a few songs I WISH were still in print so I can buy a new copy, but they’re not. Copyrights prevent someone else from saying “hey, this song is good, people would pay for it, I’ll sell it to them since they can’t get it anywhere else.” This is what Rhino Records began doing. They’d get old songs, get the rights to reprint them, and sell compilations.