I understand this is possible
Maybe not as easy as you think. Multi-channel recording (over 2) in Audacity can be done, but the interface to the computer is not simple or ordinary.
The two normal sound channels in a computer, Record and Play, are both stereo (2 track), and to get numbers over that requires special software, drivers, and configurations.
We publish a separate forums for multi-track. Some of those posts send you to other related posts.
Audacity doesn’t play multi-track, I believe ever.
Rather than go straight for three or four channel multi-track, you may be a candidate for overdubbing.
Lay down a backing track and then play that back while you perform the next track. Play both of those back… and keep going. You can be three guitars and four voices. Ever sing harmony to yourself?
This has the advantage (other than not going over stereo) of giving you “clean” tracks. Only guitar and only voice. The performer in your video comments right at the top that he’s going to get vocal spill on the guitar microphones. He’s stuck with that, but if you overdub, you have completely independent tracks and you can mix them together as you wish.
For really real multi-track it’s not a bad idea to not do it on the computer. Scan through the forum of people having trouble recording a single voice track much less three or four music tracks.
There are multi-track recorders who’s day job is multi-track, not a computer being forced into it.
On the other hand, and I have done this once, produce your whole multi-track song in the mixer, and deliver stereo to the computer. You can easily do that all day long with relatively simple and affordable kit. The down side, of course is you can’t remix it. If you want your voice a little louder, so sorry. Sing it again.
You can also, as the obsessive engineer would say (raising hand) use two computers. Put two sound channels on one and two on the other. Shuffle files around later.