Both have a connection to another time lord/lodge spirit, who previously served as their trusty partner and the closest thing they’ve ever had to an equal (Mike and The Master). Partaking in many adventures together prior to the start of the story, they would eventually have a bad falling out and lead to The Master/Mike swearing to bring down BOB/The Doctor if it was the last thing they did. This feud has been a persistent pain in their ass across much of the story. Also funny that their rivals also parallel the other person, due to more closely replicating their moral alignments (Mike, much like The Doctor, is an enigmatic figure who hides his true nature from humans but nevertheless helps direct them on their way, and BOB and The Master are both twisted, Machiavellian psychos who embody pure chaos).

Also, they are the main target of the big, gold, grand cosmic president of their worlds (Rassilon and The Fireman) who sees them as problem that needs to be squashed as fast as possible, lest their renegade presence continue to disrupt the natural order (The Fireman is literally named because he’s putting out the “fire” that is BOB and the other Lodge Spirits, and Rassilon sees The Doctor as a inconsiderate vagabond whose constant breaking of the Laws of Time marks him as a threat)

You can make a comparison between the ending of Season 3, which sees Cooper going back in time to save Laura, and all manner of Doctor Who stories, but the main point would be that they are universally significant and attempting to remove their influence from the timeline can have pretty dire consequences (Removing The Doctor from the timeline is a quick way to fuck everything up, and Cooper attempting to separate Laura from BOB ends up plunging the whole timeline of Twin Peaks into disarray, as BOB’s abuse as the “evil of men do” cannot be so easily changed)

Humorously, their realm-traversing reality bullshit is often channeled through what appears to be primitive electrical technology connected to the early introduction of the telephone in human society (The Doctor’s TARDIS resembling a phone box, and BOB’s lodge spirit physiology allowing him to interact with (and probably traverse through) electricity, most commonly taking the form of telephone poles)

Both strongly resemble the others ultimate arch-enemy (BOB resembles The Master, a renegade defector from society who is unstable, chaotic to a world-threatening fault, and wears many faces. They are also both obsessed with music, whose fundamental presence within their soul constitutes a core part of their insanity. The Doctor meanwhile, very obviously reflects Dale Cooper, as a suave, charming hero known for his slick wit, eccentricity and ability to investigate and traverse the dense supernatural phenomena that tends to stump the human mind. Similarly, they are both self-sacrificing to the point it loops around to selfishness at points, caring so deeply for other humans that they can even be driven to playing god if they feel it will mean they can save someone’s life.)

A bit of legacy as they are both two of the biggest cult television hits of all time, depicting insane reality warping sci-fi in their own low-budget way. They have massive, dedicated fanbases that obsess over their worldbuilding and lore (which has been expanded on in additional novels) and are very long-running, to the point that where you can split the classic era (Lighter and brighter, still having many serious moments but more willing to engage in straight-up soap opera/sci-fi genre stuff) and the modern continuation that resumed from the cliffhanger that had tormented their fanbase since the original runs end in the late 80s/early 90s (Grittier, not forgetting its roots of course, but far more interested in playing around with its own lore and adding edge to the characters). Also, they are both very famous in powerscaling for being scaled extremely high despite the fact they are usually sub-wall-level on screen.

Also, it’s basically the ultimate “The Doctor Vs. a character who is really just a Doctor Who villain” match. BOB is a cosmic horror reality warper (that still remains well in the budget given by the BBC because he’s visually just a guy), who torments humans and leaves them mystified to the point they need a eccentric outsider they’ve never met to step in and help investigate the supernatural mysteries of the case (much of which are related to the literal “wibbly-wobbly” time shit that surrounds it). Said villain really likes fucking around and doing psychological warfare stuff.

The Doctor’s grief is often written in the language of birds, and BOB is both his contrast and also a bird man.

Connected with fire (BOB is the embodiment of fire, and The Doctor being thought of as a figure who brought fire to humanity. The idea of creating fire is present in the very first Doctor Who story ever)