Privacy

If I buy a computer on Amazon from a trusted brand like Apple or Lenovo, will they keep my name and address and connect that information to the serial numbers of the computer's parts?

Most retailers don't do this - (edit: specifically they don't correlate a laptop hardware ID like the MAC address with the purchaser. Not because they can't, but because they don't need to. They know most users will log into something that tracks them minutes after they get home. That said yeah, they keep a record of the serial number. But the serial number isn't particularly useful for tracking your online use of the device.)

I'm very wary that Amazon has the most motive and ability to do this (correlate your network chip MAC to your real name) and probably wouldn't tell us if they did start, until long after. Followed by Google in a close second.

That said, a Chromebook accomplishes every kind of tracking the moment I sign into the mandatory Google account, anyway.

Same with any Windows laptop and Microsoft, today.

Edit 2: Lots of devices support rotating network MAC addresses, now, anyway, at least on Android and Linux. If you're concerned, it's worth looking into and setting up, as well.

Would it be wiser to purchase it from a physical store using cash, or a second-hand computer from ebay instead?

Yes.