Amazon does not register hardware with Apple, nor does any other vendor. Neither does a customer purchasing the hardware need to register it with Apple.
The serial number search provided by Apple Support, as far as I am aware, provides no results for peripherals. It does provide results for those items such as an iOS device or a Mac with which you would sign into iCloud. The act of setting up the device and signing into iCloud has the same effect as registering a product, which some manufacturers still offer but which Apple cased quite a while ago.
I recently bought an iMac with three peripherals: a Magic Mouse, a Magic Keyboard, and a Magic Trackpad. To answer your question I located the serial number of each of devices and entered in the Apple Support serial number search box. For each search I was informed that the serial number was not recognised. The iMac was recognised. This is what I expected, because there wasn't a way for the serial numbers of perhiperals to be lodged with Apple as part of my setting up the iMac when I received it (and it was bought directly from Apple on the Apple Online Store).
Keep your sales receipt if you need to return the Magic Mouse for repair or replacement under warranty in the next year. The serial number of the Magic Mouse will also be on the underside of the device.
The act of registering or not registering a purchase with a manufacturer does not affect your consumer rights under warranty.
Useful links regarding finding serial numbers are:
Find the serial number of your Apple product
Find the serial number of your Magic Keyboard, Magic Mouse, or Magic Trackpad
Find information about your warranty or AppleCare plan
With regards to your concerns about the mouse being a genuine Apple item, you could have a look at it in your Mac's System Information. Use the About This Mac option (under the little Apple logo in the left of the tool bar), then More Info > System Report.
Once the system report is generated click on Bluetooth. Scan down until you see Magic Mouse. The Vendor ID for it should be 0x004C which is the code for Apple, at least according to this source.