Could this be explained by a less-than-ideal recording setup or incorrect settings? I recall seeing a post in the last week or two that mentioned something about how recording a certain way through USB (maybe in mono) could make the result sound 2-D or lifeless. I can't seem to find that post right now.
I'm actually in a similar situation. I use an Alto TS112A as a monitor when playing out and at home. It's not the best solution out there, but it does the job for me and I still get inspired by the tones I hear.
When I record through USB (in mono or in stereo) in Reaper, the result is nowhere near as full, dynamic, and rich as what I hear through the Alto. (I know in-the-room tone is going to be different, but this is a nearly-FR nearly-FR speaker reproducing the tone, not a guitar cab that suddenly sounds much different when mic'ed.) My recorded tones sound nasal, honky, shrill, and lifeless, and there is far less low end than what I hear through the Alto (or through wedge monitors onstage when I play out, or a different brand of PA speakers at a friend's house). It sounds uninspiring, like I'm playing through a cheaper modeling unit. This is the case when I listen back using Sennheiser HD280s, a Sony stereo receiver into Bose bookshelf speakers, or several different pairs of earbuds. While these options are not the most ideal, when I use them to listen to demos or tutorials with unprocessed, direct-out audio (like Bulb's or Mikko's or Burg's), they are capable of conveying full, warm, pleasing tones with dynamics and punch and clarity, so it seems like my recorded tones should be capable of sounding as least as good as the demos through the same devices.
I'll have to experiment with using the headphone out and comparing that to the recorded result to see if there's a difference or if I'm relying too heavily on the Alto for my reference, but like I said, the Alto sounds like what I hear through wedges and other PA equipment.
Here's an example recording, USB direct into Reaper:
Through the Alto this preset sounds fat and bouncy and creamy, and has the dynamics and feel and squish of a cranked amp, but in the recording it's pretty dull and harsh and uninspiring. When I first recorded it I felt like it sounded decent through my Bose speakers, but as I listen with earbuds or the HD280s or even the Boses with a more critical ear, it's kind of disappointing.
When I record in stereo I'm selecting the track in Reaper and changing the "input" setting from mono to stereo. Should I be panning something? In my preset I use a stereo cab with L and R panned respectively, but I think I'm either summing L+R in the Axe or maybe "Copy L to R". My preset goes AMP -> CAB -> REVERB -> OUTPUT. Should I change a stereo setting on the Axe? Is something in the reverb block or later destructively interfering or losing stereo information?
[EDIT: Another recording I did with similar results used the UltraRes sample 4x12 Vox Beatle IR in mono, so I don't think the Axe stereo settings are the cause of this issue.]
In any case, in the video in OP's post, Misha pulls up a default amp model with settings at noon into a specific cab model and records direct out, eliminating all of the variables like advanced settings, FRFR choice, in-the-room sound, etc. It seems like someone (like OP) with a similar guitar with the same wood, same pickups, playing into the same amp model with the same settings and with the same cab model should expect to hear a result maybe 90% similar to what's in the video. I think his recording sounds maybe 60 or 70% similar.
Can pickup height, a master-built guitar, pick type, and other relatively small details really account for ~40% of the final tone? At around 1:15 in the video you can see that he's strumming with a somewhat loose wrist, not attacking the strings or anything, and he's playing simple chords and palm muting, not anything that would be in the realm of "tone is in the fingers."