This is an attempt at collecting, preserving and organizing and elaborating on mine's and @good-to-drive ideas and points about the similarities between Wings and The Beatles, and the commonalities between these fandoms. So to not let those ideas languish in some random comments section, I tried some write up.
NOTE: I - overall - do not differentiate between their and my points.
I think we can safely say we are living in a Wings revisiting, Paul glazing and John discrediting time. I, similarly to good-to-drive, believe that the Paul glazing and the attempts of considering him the "genius" of The Beatles have to do more with a backlash against John than some genuine belief in his musical or contribution superiority.
That being said, there are similarities between Wings and The Beatles, which can also explain why people would consider Wings (or Paul) to be the spiritual successor to The Beatles - alias why Paul was able to recapture the same audience - therefore why a Wings fan is almost always a Beatles fan, while John solo music is regarded a little differently (either in positive or negative way).
Wings featuring prominent vocal harmonies. This will be always my first point and a particular hill I'm dying on. Wings (and even Ram) having vocal harmonies means we already have the most recognizable Beatles trait carried over into Paul's solo stuff, that John and George don't really have. I - assume - this is the main reason why John's (and George's music) can feel to some as more "boring" or flat: they lack a dimension in production which would feel familiar, the vocal part can feel less prominent and dynamic, the music either feels more spacey or more prominent. Fathom great video really shows how essential harmonies are to Wings.
The band dynamic/environment. John and George still worked together with other people - so they did not record anything by themselves and also let those musician stretch a little bit - but that's still different than working on songs in band environment, especially if Paul took those people into tour with him. Even if Paul was the main writer and producer, his songs could still naturally evolve. That being said, this is a point where I'm a little shaky on, because Paul is not that great of a team worker - even when he tries - and you can actually hear with John, especially on the Walls and Bridges sessions, how he fostered the environment and how the songs were shaped. I actually think Walls and Bridges is the John album that was the most influenced by playing with naturally with the session musicians. I think All Thing Must Pass also can be put into this category.
Building up from the ground. Similarly to Hamburg era Beatles, Wings was built with the spirit of "let's get out, play to these random people, at random "venues" and play cheap." Paul had a willingness to gown down to the "people's level." We could even say Wings carried on the pre-Revolver Beatles legacy; writing songs that they were actually able to play, without getting into those pesky Studio Techniques.
Touring. Naturally, connecting to the previous point, Paul also toured. John had like one concert and some on-off performances and George had one tour. Paul was on the road consistently, he went to people, he "advertised" himself this way, it made him feel more "down to earth." John and George were basically doing the later Beatles strategy, of millionaire musicians doing hobby music releases "from comfort." Paul gave people what they wanted, fans like to have events instead of distant musicians gracing them with their presence at every blue moon. Paul also threw in some Beatles songs after the whole dissolution, and a little fanservice never hurt an artist.
The whimsy. 70s rock journalism did not like the whimsy - it did not feel like "serious" art, but whimsy (and psychedelic songs) were an important part of The Beatles in the 60s. And John Lennon - who was arguably the most whimsical Beatle - mostly get rid of his, aside from some ridiculous or nonsensical elements (the "cookie" in Hold on, Old Dirt Road, #9 dream, Mucho Mungo). That meant that for a short period of time in the early 70s, John was regarded better critically than Paul, who continued on with the Nonsense and Whimsy with full force, despite the changing times. What is Mary had a little lamb but Yellow Submarine all over again? And what does the critics matter when people want whimsey? Especially now. Taste changes over time, it goes away, it comes back. And to the cynical and critical modern eye, whimsey and fun can feel better, more honest, safe and PC friendly, than the sometimes clumsy and faulty, albeit earnest politicking of Serious Lennon.
"Granny songs." This is my new point. John may have bashed them, (which bashing critics and certain fans imitated) but these old timey tunes and english stuff were done by Paul in The Beatles AND after The Beatles. I, a once Evil Rocker, smile if I hear one - it feels so british and so deeply Paul. They are cute!
The personal experiences of a "complicated man." This is definetly a point that I personally would never begrudge people for. While I believe the best art is done by complicated people who has (had) some tendency for darkness, the actual reality behind sometimes touching, beautiful or painful lyrics still can feel horrible. For example, I consider Jealous guy to be a deeply important song and lyrics, a very real confession about a very real issue, but I understand if it feel little wacky from a physically & emotionally abusive man. Maybe you feel he is manipulating you to feel sorry for him! I get it. This one really ties back to the John backlash and the very real pain he caused, especially because we are having more and more awareness of abuse and domestic violence. John personality is woven into his music and lyrics, and while I think his character is interesting and his lyrics and music are good (and because he is dead, you can listen to it without guilt) it doesn't change the fact that it was a very real darkness and complexity that caused real pain. Def less fun than Uncle Albert/Admiral Halsey...
A more Beatles consistent persona & branding. Paul had a more consistent outwardly facing persona - while the setting and context changed and he brought in the Wife Guy and the Farm Guy elements, but he held onto the wholesome, aw-shucks charm he was selling with the Beatles.
More nostalgia/connection towards The Beatles. While Paul also expressed - at least retroactively - that he had desire in the 70s to leave The Beatles behind, he still carried less baggage towards it's music and production than John and George. Paul did really have bad experiences in working on songs in the studio and with the producers, he liked the end results. George wanted to be different - his songs were passed over a lot, he was dismissed by Emerick & Martin, his guitar playing was made irrelevant in the light of Paul just doing anything. He also wanted place for his faith in music. John - also in the light of him just changing his mind all the time - was also treated sometimes as somebody who is stupid, whose songs were maybe taken into direction he did not want yet he wasn't feeling in control, to be able to get it back from Paul, from George Martin etc. The White album was the first time John really get behind the wheel for most of his songs - probably thanks to Yoko, also incorporating screaming vocals and self-expressions, inspired by Yoko, and using from this point onwards. Paul having more positive memories concerning recording Beatles songs, was more inclined to built on techniques and sensibilities he learned there.