The short answer is: not really, certainly not to the level of detail that we have in cars/computers now. Long answer below.
What we know definitely did exist at the time was distinctions in the quality of weapons/armour. A suit of chain mail could, for example, be labeled as 'best' chain mail marking it out as exceptionally good. We see similar information with regards to arrows, longbows, and other weapons (I'm most familiar with the archery/armour examples, though). These labels survive to use primarily in the form of details from economic transactions rather than Knights discussing their weapons. 'Best' or 'Good' quality weapons/armour cost more than those whose quality we presume was just 'normal' (they are not usually labeled as anything in particular). There are also instances of royal records, such as under Henry V, demanding X number of arrows, of which Y must be 'good' quality, for a given campaign.
In general it is difficult for us to know what the exact differences were between normal and 'good' quality. We assume that the higher quality items were made of higher quality steel. However, it is not clear if the stated quality of an item actually matched up to reality. Medieval smiths did not have a lot of tools to test the quality of the steel they made so they simply had to guess, based on their years of experience in fairness, how good the armour they made was. We expect they were pretty good at this, but we can't be sure. Then both he and any merchant selling the item would have to honestly represent the quality of the item, which is harder to determine the frequency of.
There is some reason to guess that this system did not entirely work. In the Later Middle Ages there was a system in plate armour used by some smiths where they would put a number of makers marks related to the hardness of the armour being sold. Two makers marks signified very strong armour, One mark was noticeably weaker. This is sometimes reported in the form of how well they would stop/deflect an arrow from a certain distance. The problem with this system is that when Alan Williams tested the metallurgy of several suits with these kinds of marks he found no noticeable difference based on the number of Makers Marks the suite of armour had.
There would likely have been an element of stylistic armour choice being made by knights, especially in the mid-late fifteenth century when two very different styles of plate armour (German and Italian) emerged. However, these would primarily have been full suits of armour, not individual specs. As a crude comparison, it would be more like preferring a BMW to a Mercedes rather than discussing what engine you have in your car.
I should probably point out that I'm more of a technology person than a Knight person so someone else will have to provide a detailed answer on whether knights discussed the quality of their weapons. I expect some practical discussion to have happened (otherwise there would be little point in the distinction of 'good' quality weapons in written accounts) but I would not expect it to have been anywhere near the level of detail that car buffs obsess over. I'm happy to be proven wrong, though. I have also almost certainly forgotten something...so hopefully someone will come along to remind me! :)
Claude Blair's European Armour c.1066-c.1700 is, in my opinion, still one of the best books on the development of armour in the Middle Ages.
Mathew Strickland has a good discussion of bow/arrow and armour quality (incl. the Maker's Mark thing) in The Great Warbow
Alan Williams' staggeringly dense book The Knight and the Blast Furnace has a really extensive discussion of medieval armour quality, but is hardly an approachable book. He's written some more approachable articles I can provide references for if you're interested, though, but I'll have to dig the references up first.
Not anywhere near what we have today. Nowhere close.
Point A: There's some very complex things that need to be in place that allow for any kind of authentic specifications akin to what we know today. For example:
These are all products of the industrial revolution and the modern age, when machining grew up to the point of being consistent and reliable. When talking about "Medieval" anything, you're talking about hand-made objects. Hand made objects, while capable of great quality, will never be consistent enough to allow for "specs" in the modern sense of the word. So there was no way to establish an agreed upon system of analyzing and categorizing an objects performance during the medieval time period. The scientific method literally did not exist.
Point B: Our culture of obsessively categorizing objects is mostly post-victorian. When you read sources such as medieval treatises and books on combat training, one of the most striking things about the language is how vague it seems to us in the modern age. We think of "bastard sword" "long sword" "flamberge" "two handed sword" "espada" "foil" "rapier" "viking sword" "Oakeshotte type WHATEVER" etc etc. But these historical texts written by Meyer or Liechtenauer and others refer to things as "sword" or "big sword", and that's pretty much it. This is a constant point of endless contention and debate within the HEMA and scholarly circles, and I think it's fairly obvious that it inhibits the idea that people were able to adequately "compare specs" during this period, because there is no consistent categorization. Imagine how worthless an opinion would be when comparing which car was "better": a 1960's muscle car or a rock-crawling Jeep. They're both technically cars, just like everything the texts refer to are technically swords, but there's a huge degree of variation that isn't accounted for. While this doesn't strictly make comparison impossible, it obviously makes it more difficult. Not to mention the huge variation in languages, and the very poor level of general education including literacy.
Point C: Disposable income. Man people are incredibly rich these days compared to back then. Most men who were professional soldiers during the medieval period were not mail ordering their "kit" one piece at a time. They would be measured and have the equipment custom made for them (if they were rich enough) or they would be wearing lesser quality (padded jack alone, gambeson only, leather coats, etc) gear as they could scavange/afford or be equipped en masse by their lord (aka munitions plate). The point is you didn't have a bunch of gear queens comparing online deals for the latest tacticool punisher themed drop leg holster or blowing 7k on an AR that's never going to shoot anything other than paper.
Concession: Human nature is one of competition. I'm sure people have compared quality and performance since before written language was a thing. In fact during the pike and shot period armor makers would often shoot their breastplates with a pistol, to show that their gear was so good it was bullet proof. Google up "proofing marks" on armor for more info there. So yeah I'm sure they compared things, we see folk lore littered with mythical items (Hello Excalibur) and fine quality weapons and armor being so finely crafted that they are works of art in their own right. But compared with "specs"? Compared with obsessive measurements and performance statistics across multiple weather conditions? Accounting for error? The scientific method? Not a chance.
To add and act as a supplement to Valkine's excellent answer, consider the manufacturing requirements of the scenario you prepose; for a comparison to be reasonable you have to have a degree of reliability in part manufacturing that's not obtainable in hand crafted goods. This wasn't really possible until the advent of factory assembly and standards of quality control that would allow for interchangeable parts.
Yes! At least in the late medieval and renaissance eras.
There were grades or categories of armour, and indicators of such for the purchaser to inspect for their satisfaction. Mail was made and sold in several grades, some easily observed (eg. single vs. double-riveted, the weave, the thickness of the links), and some harder to know (eg. the temper of the links).
The general concern for both buyer and seller was what sort of blow the armour could take without rupturing, and it would be tested (proved) against just a hit. The proof was effectively the spec, and there were many specific proofs you could get, all the way to bullet-proof. There was a "toute epreuve" (sorry, accents are hard on this keyboard) which covered both windlass crossbow and sword, and a "demi epreuve" which covered level crossbow and thrown javelin, and various arrow proofs, since there were various arrows that could be encountered on the battlefield.
The certification (proof mark) was stamped directly into the steel of armour plates, sometimes into each piece individually. These proof marks were often the actual impact dents of the certification test - in the case of bullet proofing, an actual bullet dent from a fired gun, which would sometimes be embellished and decorated to highlight it. Charles ffoulkes recounts an anecdote from 1590 where an Englishman wished to have armour of English steel instead of the conventional German steel; he insisted on it being made at the same weight as the German armour, and subjected to the same proofing process. The German armour took a dent, but the English armour was shot clean through, and ripped open for several inches. So the certification process certainly worked in this case.
Front and back pieces of armour were proofed to different specs, as the front was more likely to receive direct fire, whereas the back was more likely to get hit by random spent bullets, and was a good place to cut down on weight. The stronger piece would be musket proof, while the weaker were pistol proof. Helmets were similarly rated to musket proof or sword proof.
The bullet weight and amount of powder used in a proof was obviously of great importance to all, and it was sometimes proved by the buyer before they paid, if they doubted the armourer's claims. Some armourers were known to have refused second proofs of fine armour, because it would spoil the artisanry that was applied after the initial proof and facing work.
Lastly, the armourer's mark itself acted as a form of certification. In the guild era, reputation of craftsmen was taken so seriously, that faked ("forged") armourers marks were not unknown to capitalize on others' good reputations (this was more common among weapons than armour). Basically, once an armourer's work was generally known to be proof against the blows he claimed, the armourers' mark was itself all the certification the buyer needed.
Source: The Proof of Armour, in The Armourer and his Craft, Charles ffoulkes. Originally published by Methuen & Co, London, 1912.