The sound on the video at that point is not related to the sound that mic is sending into the DAW.  That would be top quality.  The sound of the video is the sound of the camera mic I would say and is a good indicator of how that room might sound.
 
Recording even in a tricky environment like that with a half decent mic say 3 to 4" away with a pop filter in between will produce a result very similar to a studio environment done a little further.  In less than ideal rooms just get much closer on the mic.  (the pop filter becomes more important) If the performance is good the end result will be pretty good as well.  You definitely don't need any portable vocal booths on tour.  There will always be a spot in the room that sounds good.
 
The most important aspect is the fact those ideas were being captured at that very time.
 
Many reasonable hotel rooms are going to be well furnished and sound quite dead.  As long as there is not too much interference from adjacent rooms then decent acoustic guitar tracks could be laid down as well.  The convolution reverbs can add some slight room behind later on.
 
The idea of recording while on tour is really good one.  It is a time when everyone is very focused on the music and naturally other musical ideas are going to emerge on a tour.  This is one area that our current technology has really changed this aspect of recording away from the studio.  Just add in a powerful hardware synth or even just a controller and you can add tons more material without disturbing anyone.

With regards to that video, I didn't watch the whole thing but are you absolutely sure they aren't just recording a rough demo? 

One great singer here (I forgot his name), ALWAYS sang in his sister's bedroom ... facing the long window curtain (about 4-5 feet away from it, iirc).
 
Early reflections near the mic are one of my biggest enemies (the midi-synth, desk, heater, etc.).  Such *verbs* cause comb-filtering or something.  So you/I remember to just diffuse those hard surfaces near your mic ... with a jacket or sweater or blanket.  Allow 'some' of the rooms imperfections to 'color' your or another's vocals ... much sweeter than that dry-but-unnatural Real-Trap's-portable-vocal-booth (imho).
 
I am in a hotel now and ALWAYS carry a $100 dynamic mic (as others stated above) ... for when the muse awakens.  I will sing into the blessed long curtain (rarely into the wall corner (as others successfully do)!).  Every vocal performance takes me about 40 takes anyway (even after much warm-up) ... oft with different mics ... different environments ... and different ambients ... in every 'painted' song.  
 
But the majority of my 'imperfect' vocals become more 'manageable' (with comps, melodyne, verb, etc.) if condenser mics in a padded studio are used (not by Bat's impeccable logic).  Dynamic (hotel) mics require extremely inspired (perfect) performance-takes to become acceptable in the long haul.  Perhaps they sometimes 'overpower' the song-painting (if that makes sense) ... especially if the song painting has competing rhythmic enticements.  The talent must dance with the rhythm or dance with the magic chords to unite the painting.
 
IOWs, overly dynamic vocal performances ... seemingly can destroy a song-painting that relies on rhythm as the dominant element.  (if that makes any sense at all  :):):) )  I dunno.