Ok, so first of all, it's ok to be confused - this is a confusing topic and requires ditching a lot of very deeply subconscious ideas about how the medieval world worked. One of the problems is that, no we're (me and you/historians and the general public) are not on the same page. Let's break that down point by point:

Did gradually emerge

Yes, the socio-economic structure of medieval Europe emerged and evolved gradually - there was no top-down imposition.

From an early somewhat informal/arbitrary stage all the way to a later formalised/systematic stage

Yes, the socio-economic structure of medieval Europe became increasingly formalized. The problem with that statement, however, is that it's so vague and general that it tells us nothing, because everything became more formalized as the middle ages went on - law, politics, literature, education, religion etc etc - and has continued to become more formalized, more or less, to the modern day.

From the post-Roman-Empire Barbaric Kingdoms all the way to at least the second half of the 15th century

Yes, but again this doesn't really tell us anything. History always goes only in one direction!

In at least but not limited to most of the European West (let's say today's England, France, Germany, Northern Italy, Spain)

Nope, not at all. Even these core regions of western Europe were profoundly different in the middle ages and the social/economic/tenurial structures were equally varied. We can't even reasonably talk about 'France' or 'England' in this sense - the differences between north and south in both those kingdoms were so drastic it's difficult to generalise easily about them.

A system, that we call with the word "feudalism"

The problem here is less with the word 'feudalism' but the word 'system', which implies some sort of uniformity or design. Like I said, the periods and geographical regions of Europe were so varied it's silly to talk about a 'system', when no part of the system fits with any other part.

That created a common social, economical and political landscape/background

Again, no. There was a common mental landscape in medieval Europe but this was created above all by the institution of the Latin church and the intellectual legacy of Rome - it was a culture created in spite of, not because of, the socio-economic structure.

Based on the rewarded (e.g. privileges) and exacting (e.g. duties) personal allegiance/loyalty/service between people?

Again, this one is kinda true - medieval aristocrats valued loyalty and service, believed the world was essentially hierarchical and rewards were expected for these things - but the same could be said for virtually any society. I am rewarded (e.g. pay and leftover chicken nuggets ) for exactions and duties (e.g. scrubbing the fast food restaurant's toilets) created by the personal relationship I have with my manager - does this mean the 21st century is feudal? It's another statement which is so bland that's it kinda pointless.

The point to take from all this is that the social and economic structures of medieval Europe were so diverse and localised that any model of 'feudalism' which is proposed either ends up being so broad as to tell us nothing (as the one you described would be) or so specific that it only applies to one region (which kinda defeats the point of having a general model).

Now, none of this means that we need to ditch the word 'feudal' altogether and certainly doesn't mean that we give up on trying to give a general picture of what medieval society looked like. One of my biggest gripes with the whole 'f-word' crowd ("don't say the f-word!") is that it never provides an alternative view of medieval society, it just tells amateurs enthusiasts and undergrads to forget everything they already know. The reason feudalism is so persistent is that it neatly describes what medieval society was like - even if it's fiction, it's difficult to dislodge it if you aren't going to give people the information they want. We need to continue to attempt to build models and write histories of medieval society which can be easily pictured and understood outside of scholars of the middle ages.

The problem with doing this is that it's really fucking hard. In the 20 years since Susan Reynolds utterly eviscerated any last remnants of people supporting the old view of feudalism, not one historian has written something aimed at explaining medieval society to a general or even undergraduate audience (notable mentions to Chris Wickham and Fred Cheyette, both of whom kinda tried in different contexts, without really tackling the problem head-on). Medieval Europe is so varied, it's very difficult to say anything without lots of caveats. It's easy to explain how it worked in particular contexts (11th century France, 15th century England) but much harder to generalise.

Rather than try to do so myself now (I'm tempted, despite the fact it merits a whole book...), I'll just leave it there for now. The original question was about the historiographical implications of the 'feudalism isn't real' movement, so I'll stick to that. If there are further questions though, about anything I've said here or about the general contours of medieval society, I'll be happy to give them a shot.

Has modern scholarship challenged the idea of serfdom? Or to be more precise that many / most farmers were legally tied to landowners or the land to which they lived, and that were expected to work a certain amount of time ploughing the fields held by the Lord in demesne?

Since apparently Feudalism never existed, does that mean that "Personal" politics weren't ever a thing?

By personal politics I mean politics dominated by the bonds different men had to each other (For example, in a society dominated by personal politics the reason the Duchy of Burgundy would be part of France is not because it is so by law but because the Duke of Burgundy swore loyalty to the King of France or something, and the reason the Duchy of Burgundy is big is because the different lords of the region known as Burgundy are loyal to the lord who calls himself Duke of Burgundy etc.) Sorry if that's too confusing, I'm kinda drunk right now

Hi there!

There is more to be said here but make sure to check out the AMA /u/idjet did on the subject of Feudalism.

Thank you!

Also, I had always assumed that there were a lot of commonalities in the political and economic structures of the core areas of European Catholic Christendom excluding Italy which always maintained a semblance of civil bureaucratic authority. While there was a lot of diversity, there are certain features that made these regions distinct from other areas?

As a follow-up question, what does current scholarship say about economic feudalism, IE the economic system before (continental) Europe adopted capitalism during the second half of the 18th century?