There is one simple fact: New unused logic boards that are genuine Apple parts cannot possibly come from any other source than Apple directly. That's because Apple does not sell these new parts to anyone. Even authorized service providers can't buy those and use them however they want instead each logic board needs to be assigned to a specific repair for a specific device.
Hence, this brand new logic board off ebay could not have come from Apple or an AASP.
This leaves only two options: Either this logic board isn't a genuine Apple part or it is a genuine part that was never meant to be sold on Ebay which means Apple can (and they do!) blacklist the part.
The only way to get a functional part like this off Ebay is if it came out of a working new Mac. Which wouldn't make any sense, nobody takes apart a brand new Mac in perfect working condition to sell it part by part instead of selling the entire Mac.
So now you know this isn't a part that came from another Mac (then it would be in used condition and not brand new) nor is it a genuine part sold through Apple. I know where this logic board likely came from and I know you have been scammed. Or rather, you fell for the typical replacement parts scam: Someone sells you a part that is technically as described and technically is brand new and in working condition. These sellers are careful not to make any claims about compatibility or about how to install and initialize it.
Why is that? Because they know they are selling you a part that you cannot install and initialize. For example, this scam was done with iPhone displays. These displays have the actual display panel with a digitizer above it that registers input and then on top comes the glass. You cannot replace any one of the three individually yet you can find the parts for sale. So if the screen still works but the touch is fried you might assume you can keep your original display panel and just replace the faulty digitizer but this repair is virtually impossible. You can actually replace the top glass if it's smashed in but even so you need the tools and skill to do it and you can rest assured any customer buying this off Ebay has neither.
Makes me wonder how a "new" / never-used logic board would turn up on Ebay.
It was at some point an original part from Apple's supplier but either failed QC or did not end up making it to Apple for another reason. Often these components are meant to be destroyed and then make their way to the States through illegal side channels. Because it ended up outside the proper channels it cannot be considered an OEM part as that would require Apple to be in possession of it. It would be illegal to import such a part as it is a counterfeit violating Apple IP. All it takes is looking underneath the heatsink where you'll find that M3 SoC with the Apple logo. If that's even on there. There is no telling what the state of the components is and if this board is in working condition at all.
Yet usually these boards do have the components and are simply missing the Apple pairing that allows them to initialize. Problem is that this part will never ever be authorized by Apple to initialize which is why OP receives this error message on the other Mac telling him exactly that. The only part Apple will authorize is one either bought through Apple self service repair or installed by Apple themselves.
Starting with M1 Apple has made it impossible to simply swap out logic boards and especially on Macbooks you cannot even swap ogic boards between two identical Macbooks as they now require authorization to pair with the internal Macbook display or you simply won't have a working internal display.
If OP had asked about this topic before spending money on a part we know can't ever work we could have saved him the trouble. This is all public information and many years down the road since the release of M1 this is common knowledge for qualified third party Apple repair businesses.
