Feudalism among MechWarriors is not necessarily unreasonable, although it depends how it is achieved. Remember, in the setting neo-feudalism emerged over time as a method to organize and administer unimaginably large realms. I argue owning and piloting a 'Mech developed not as a signifier of power, rather cause and effect should be reversed here: the ability to pilot a 'Mech, ASF, etc is so exceptional it is the reason for receiving the feudal title, we are just centuries too late to see this transition.
One factor that has been alluded to but is really important to 'Mech ownership is the ability to actually pilot a 'Mech in combat. The ability to pilot a 'Mech is not very exceptional, but piloting a 'Mech well enough to not be a liability under live fire, with the ability to endure concussive injuries, heat, and grueling amounts of time in the cockpit, is very rare. Remember, the story and game almost exclusively involves characters who are important in BattleTech, mostly MechWarriors. Being a MechWarrior is a rare skill, and one possessed even by the most incompetent Lyran social general.
Consider in ~3060 the size of the Inner Sphere. There are approximately 50,000 BattleMechs in House armies (5 Houses, ~ 80 Regiments each). Between students in academies, trainees/squires, particularly skilled technicians, test pilots, mercenaries, dispossessed, retired veterans, etc., I would argue a generous upper limit for the number of people able to pilot a BattleMech in combat is 500,000 in the Inner Sphere total, whether they are Piloting/Gunnery 0/0 or 7/7. Everyone else does not even qualify for PS/GS. There are ~2000 or so planets in the Inner Sphere. Not every planet has 1+ billion people, but I think a mode of 10,000,000 inhabitants for most planets is reasonable. Throw in a few dozen planets with populations over 1-10 billion people, and maybe a comfortable estimate for the size of the Inner Sphere approaches 1 trillion people. Honestly, this feels like a low estimate. At any rate, if we accept ~ 1 trillion human beings, that means 0.00005% of the population registers a piloting skill and gunnery skill of any skill (for comparison, being a current major league baseball player, including being able to hit a MLB pitch is a skill only 0.0001% of people are capable of doing, or 2x as easy; that number goes way up if you include other leagues, training leagues, and retired vets like we did in the MechWarrior numbers).
Bringing this back to neo-feudalism and owning a 'Mech, there are haves (with title) and have-nots (serf/peasants) who are tied to the land and insulated from combat. For the have-nots, what percentage of the population of the Human Sphere has enough free time, money, etc. to be able to leave the world of their birth regularly? How many civilians have been involved in combat since the 2nd Succession War?
As for who the haves are, historically feudal title, from sovereign to knight, is awarded for or won by (military) service, although it can also be bought by wealth or great charisma (excepting inheritance by future generations, this pertains to succession). The rarity of title means those who possess it are exceptional in a manner analogous, though not necessarily proportional, to the rarity of being able to pilot a 'Mech in combat.
In this way, developing neo-feudalism seems just as plausible as other have-havenot paradigms, and the BattleMech is as reasonable as any sci-fi knighting sword.