Does this song have a tempo or time signature ?
How do you write and play this kind of music ?
It's tough to know how this was notated without seeing a score of some sort. With that said, you do ask a very good question: how do you notate and play music like this?
Some modern music does not have rhythm or meter in the way that we expect from music by someone like Mozart. Some scores have a listing of various musical gestures separated by durations of time. Consider, for instance, this excerpt from Penderecki's "Threnody for the Victims of Hiroshima" (the video includes the score where you can see more instances of this):
Notice that's there's no meter or rhythm per se, but there is a listing on the bottom showing that the first unit should last 15 seconds and the latter unit should last only 11. In this sense, I guess we could say that the tempo of the piece is 60bpm, but since we don't sense this as a listener, that's seems like a trivial claim to make.
While this piece probably has many fermatas in its notation, and it's probably in mixed meter (if not in free time), I'd say this piece has a tempo, and it's quite slow.
I'd generally notate much of the faster piano runs as if the first note of the run is at the start of one beat and the last note of the run is at the start of the next beat. Accelerandos may sometimes need to be notated as a result.
I'll see where I can get to over the next day or two when I have time. It's an interesting challenge!
The first stage of the analysis is to note that there is a repeat at 1.11 on the video. This means that they are not just improvising in free rhythm.
Having noticed the repeat, with a bit of fiddling, you can start at 0.00 in one instance of YouTube at the same time as starting at 1.11 in another.
You will find that it is an exact repeat except for slight differences in overtones on the wind instrument (What the heck is that? It's not a flute and the notes are very 'dirty'). In fact the timing of the repeat is so precise that I suspect they are playing in time with a click track.
Conclusions so far. It's a composed piece with a straight repeat, played in metronomic rhythm.
I'll have another go tomorrow and add to this answer.
No, this song does not have a tempo
Update: This song may or may not have a tempo.
And you can safely argue that there is no time signature either or that it is irrelevant.
In a performance or rehearsal setting, you'd need a band leader, or conductor to visually/physically cue entrances or beginnings sections.