So, bought a computer from Amazon [...] Alienware [...] Less than a year of use, the thing cooks.
Let's throw the warranty thing out; there was a stalemate in terms (not in the post text, but described by OP elsewhere). The 3rd party warranty service couldn't be provided as purchased, and it was refunded. He accepted the refund. No warranty.
Amazon tells me to send the laptop into alienware for repairs.
Amazon has no obligation as the seller to service this product outside the terms of an Amazon-sold warranty. They pass the ball to the manufacturer, and as there are no warranty terms in place with them, a simple service request begins:
I contact them and explain the problem. The service tag links to the wrong system though [...] my wife has an identical laptop that links to the right hardware.
Apparently, his wife has an identical $2,500 (I know) Alienware laptop (ST#3R4), which had previously been serviced and replaced by Dell for "overheating issues" with (ST#8G0). For reasons unclear, they confused his laptop to-be-serviced (ST#393) with his wife's current laptop (ST#8G0). Strike one, Dell.
They advise me to send it in on that tag instead.
Whoops! They advised him to send in a device with a serial number not matching the one on the service ticket. Strike two, Dell.
Now, let's look at what Dell had to say for themselves after they had bungled the service ticket up and then advised him to do something that he should have never done:
[OP's Laptop] was received at DEPOT instead of [OP's Wife's Current Laptop] that is still under your possession.
Cool, cool.
[OP's Laptop] was EXG for [OP's Wife's Dead Laptop] but it was never returned to Dell.
Yikes. Dell has somehow associated his wife's old, dead laptop with his. Unless the old, dead laptop was refurbished and sold by Amazon LLC as new (pretty unlikely), and later bought by OP in an insane coincidence, this is an internal Dell issue.
[OP's Laptop] is not required to be returned to you as it had a HOLD to be returned as a previous EXG was completed on it.
By "HOLD" they mean that they
somehow thought that [OP's Wife's Dead Laptop] was never returned to them as part of an exchange for [OP's Wife's Current Laptop], and
somehow associated [OP's Wife's Dead Laptop] with [OP's Laptop] when generating and processing his service request.
And, as such, they straight-up destroyed dude's $2,500 laptop. No questions; no follow-up; no internal investigation. Nothing. Took it out back and shot it because it went down a dark bureaucratic alleyway and ended up on the wrong desk.
Unless OP is perpetuating the very scam that stuff like this is set-up to protect against, or is leaving out any details, Dell is very much in the wrong.