Well, OEM is "Original Equipment Manufacturer," that is, the company who originally created the end product. In the case of Apple, the OEM is actually Foxconn for most of the parts, as Foxconn builds the parts and pieces that go into Apple equipment, for example. Generally, the term OEM refers to the maker of the part, so an OEM keyboard would be a keyboard made by the same company that makes the keyboard in the Mac, for example. Technically, there are very few "Apple" parts at all, as Apple has OEM providers to manufacture the parts and pieces for them, which they ship to China to be assembled.
OK, with that philosophical bit aside, the question is whether or not the keyboard is exactly the same as what is in the machine. And the answer is, no way to tell until you put it in and see if it works exactly the same way. And even then, it may not be "exact" as there is no requirement for that. The keys may be slightly different in how they present the letters, or the cables may be slightly different in coloration, for example. Even among OEMs there is aways variation.
Let me give an example from the automotive industry. Let's say Ford assembled a car and in it the standard spark plug was from Autolite. That makes Autolite the OEM for spark plugs. But you can also put in Champion spark plugs made to the exact same specs and the car will run in exactly the same way. But if I'm restoring an old Ford and want to be "real" I would look for Autolite plugs because they were "OEM" at the time the car was made.
So, to answer your questions, in turn:
When they say OEM, does it mean that it's a genuine Apple product? As already discussed, Apple doesn't really have many "Apple products" in reality. They sell devices for whom the OEM is, largely Foxconn.
Or 'exactly like the original' but not Apple genuine. Again, Apple doesn't make much of the systems they have assembled and sell. So, "exactly like the original" is all you can get as Apple does not make keyboards directly.
Is it true that Apple doesn't sell their replacements parts to anybody? Apple has agreements for parts for authorized Apple resellers. But again, those parts may well have been OEM from Foxconn or some other manufacturer. Even in one system there may be multiple OEMs for the parts. Consider that some iPhones have Qualcomm modems and other have Intel modems, for example.
And so if anybody claims to have Apple original parts, unless they come from faulty devices (therefore used) they're lying? "Lying" is a pretty strong word. I've been very pedantic about what OEM is in this post and even I would not say that if I saw an ad for an "OEM Keyboard for a MBP" that it was a lie. For 99% purposes a part that is advertised as "OEM Apple" means that it comes from the same manufacturer that makes the part for Apple. But "100% Apple compliant" would be another way to say that a part is exactly like the one Apple ships with its systems.
In the case of your keyboard, the original link you gave says this:
OEM UK keyboard replacement for the 13" MacBook Pro Retina
Manufacturer: OEM
Apple Part Number: N/A
Base Model Compatibility: A1502
Date Compatibility: Late 2013, 2014 and 2015
Condition: New
Warranty This part is covered under our lifetime warranty, terms apply
This item is covered under our no quibble return policy, terms apply.
Parts Origin - OEM
The term OEM stands for 'Original Equipment Manufacturer', the exact meaning differs in several industries.
To us, OEM means that the part is the same part used within your device, using the same materials and specifications.
We use the term OEM to cover many parts from refurbished, used and new components, the exact condition will be stated in the items description.
I bolded what seems, to me, a critical statement. They are claiming that their keyboard is the same as the one in your Mac.
Now, does that include backlighting? You would have to ask as I don't know if the backlight is IN the keyboard or just attached to it. I suspect it is included, but there is no way to tell until you take it apart. But they do offer a full return if you don't like it, so the risk seems small to me. Smaller than going to eBay.