Just to set the record straight Microsoft Technical Support EXPLICITLY & STRAIGHTFORWARDLY recommends to use another serial number to download a required image for Surface 3 (& Surface 3 also though don't quote me for this particular model). If you so want I can post the Transcript of not one but 3 different chats with Microsoft Support representatives where they confirm and even suggest & recommend that I use a different serial no. to download a different recovery image. And this even when one of those 3 representatives had consulted his senior manager on how to obtain a Win8.1 image for those Surface users who had Win10 pre-installed. As per that senior manager too, one CAN use the serial no. of a DIFFERENT PIECE/SAMPLE of that model to get the recovery image, so you can't blame the lower-rank employees of not being aware of the do's & don't. I can absolutely provide the Transcripts of these chats (if you so want) which confirms Microsoft DOES NOT SEE ANY PROBLEM with users using a different serial number to obtain a different recovery image to what the device originally came with. Infact they didn't just suggest, but actually recommended to use a different serial no. to download a different recovery image.
I could have easily got the serial no. from a physical Service Center in my country but the thing is- Surface 3 hasn't launched in my country, only the Surface Pro 4 launched (and most likely Surface 4 non-pro also will, if & when it launches in US), which you know came with Windows 10 from day one. Thus, visiting a service center will not yield any result.
Come to think of it- the reason why Microsoft asks for your serial is to (1) make it very easy for one to download without having to go through the hassle of 'selecting' the exact image for your device, since most users are not very tech-savvy, and (2) Not force-recover using a wrong recovery image which may brick your device, and (3) reduce the download traffic just in case it is abused to bog down their servers.
The way I see it- it is an overly-automated ('overly' for us, not for the non-technical people) and probably dumbed-down, Apple-ish kind of way of doing it. Nowhere is it apparent that Microsoft does this to PREVENT users from installing a different version of Windows.
IF THAT WAS THE CASE: MICROSOFT WOULD HAVE ASKED YOU TO LOGIN THROUGH YOUR 'MICROSOFT USER ACCOUNT' WHEN GENERATING A DOWNLOAD, BUT SURPRISINGLY THEY DON'T !! ......One just has to Select the product in question, enter the serial no. and you are DONE!
This is the reason I questioned you that how exactly does sharing the serial no. affect the warranty of the particular piece in question. But apparently you are misinformed that downloading a recovery image is somehow interlinked with your Microsoft User Account. It isn't. Anyone can go, enter a serial no. and download a recovery image without bothering to login or even create a user account.