Unfortunately we can only see the front of the lens and ... not clearly enough to read any labeling.
There are several lenses that it "could" be. Are you interested in the gentle background blur?
The EF 35mm f/1.4L might be able to do that. The 50mm f/1.4 can certainly do that.
You might want to check sample images at pixel-peeper.com which will let you view sample images by lens.
I didn't see the typical wide-angle distortion. Through the video it looked like the focal length may have changed (more than one lens is probably being used.)
A 35mm lens will give a focal length that appears close to normal (you wont see wide-angle distortion but also wont get the compression of a long lens.) But as the focal length gets shorter, the ability to get the blurred background decreases and you need a very low focal ratio to pull it off (hence my suspicion that it could be a 35mm f/1.4)
BTW, the distance from lens to subject vs. distance to the background plays a big part. To pull this off on the 35mm f/1.4 you'll need a subject which is rather close and the backgroud needs to be in the distance (you don't want a background immediately behind the subject.)
The effect is very easy to pull off in long lenses -- but the angle of view gets narrow and that means you have to position the camera farther away to get everything in the frame that you want.
Here are some examples of the 35mm f/1.4 at very low focal ratio. I set the parameters for the search to require an APS-C body (he's clearly using an APS-C camera because no full-frame cameras have an articulated LCD screen) and I set the focal ratio range to only show images shot using f/2 or lower.
He's shooting into a bathroom mirror which gives us a pretty good approximate distance. I'm also noticing the labeling on the lens - which is blurred, but something is printed roughly
60º 120º apart around the front of the lens.
A lens that would match this approximate angle of view, also provide a low enough focal ratio to create some blur, and also has the printing offset at
60º 120º around the front of the lens is the Canon EF 24mm f/1.4L II USM.
I doubt anything with a shorter focal length or higher focal ratio would be able to generate the background blur. A 24mm on an APS-C body (the camera body looks like it may be a 70D - and that would make sense due to the focus system - that camera does extremely well at recording video) would be mildy wide.
Sigma also has a 24mm f/1.4 "art" lens, but that lens has no printing visible on the front of the lens -- and the lens in the video has printing. So this wouldn't be the Sigma art.
Canon doesn't make a lens below the 24mm focal length that has a focal ratio below f/2.8 (other than the EF-M 22mm f/2 STM, but this isn't an EOS-M body, so that option is out.)
This gives me a bit more confidence that this is almost certainly the EF 24mm f/1.4L II USM.
I going to assume that if this guy is using an APC-S sensor camera, then he's probably not going to be using a Canon L lens. The lens appears to use about a 77mm filter. If it's a Canon, then it could be the EF-S 10-22mm f/3.5-4.5 USM. It sells for about $600. The 10-22 lens uses the same hood as a couple of L series lenses. On a cropped sensor, this lens gives you a 16-35mm focal range, with little WA effect at the longest focal length. It also has USM, which is good for video.
The lower priced EF-S 10-18mm f/4.5-5.6 IS STM is also a possiblity, but it uses a smaller 67mm lens.
I really suspect that it is not a Canon lens. It could be a Signa lens, or one its' close cousins, that sell for around $500. The Sigma 17-50mm f/2.8 EX DC OS HSM Zoom Lens uses a 77mm filter. The Sigma Wide Angle 28mm f/1.8 EX Aspherical DG DF Macro Autofocus Lens also uses a 77mm filter. Both sell for just under $500. There are about a half dozen or so Sigmas that look like this lens that sell for under $1000.
These are just guesses. I doubt if this guy spent four digits on a lens. I could always be wrong.
The challenge with the 10-22 is that it's only f/3.5 (and only at 10mm) and at the 22mm end it's f/4.5. You'll never get that level of blur at 10mm and f/3.5 nor at 22mm and f/4.5. But you would get that level of blur at 24mm and f/1.4.
I own the 14mm f/2.8L USM II and you basically get no blur to speak of. The focal length is too short and the f/2.8 focal ratio is too high.