A close-up of one of Thom’s vintage 1960s Gibson SG guitars during Reckonerat Lollapalooza Berlin on September 11, 2016 (radiohead.com). Note that the Vibrola arm is removed.
Replicating the sound of Thom’s SG is basically the same as replicating the sound of Jonny’s Les Paul. Find any solid-body guitar with a 24-¾" scale length and a tune-o-matic bridge. Then add some vintage-style PAF humbuckers. That’s it. Then learn how to gently finger-strum like Thom on House of Cards and Reckoner. To be fair, it’s hard to overstate how important the detail of Thom’s finger-style (the way the finger crosses the strings, the picking location, etc) is to his sound on those tunes.
Thom could easily get the same sounds from Jonny’s Les Paul or from his own vintage Les Paul. But the SG is a much lighter and more comfortable guitar to play, which is probably the reason Thom prefers them, similar to his preference for short-scale basses.
You could get the Epiphone SG Standard 60s Maestro Vibrola, which is an Epiphone reissue of Thom’s vintage SG. The main issue with modern Epiphones seems to be quality control, so you’d want to try one in a store to make sure it plays ok. But if it does play well, it can certainly sound as good as a Gibson. And it can sound just like a Gibson if you swap in Gibson pickups. Of course, the resale value won’t be as stable as for a Gibson, but it’s so much cheaper that it’s hard to complain.
However, Thom has the Vibrola arm removed from his vintage SG. So there’s no need to buy a reissue with the Vibrola, unless you personally want to use it or just for aesthetic reasons.
Two of the same model solid-body guitars with the same pickups will probably still sound slightly different from each other when plugged-in (for example, two Fender Player Series Jaguar guitars). Many will suggest this is due to wood choice, finish type, guitar “quality”, etc. While those things may matter on acoustic guitars, they don’t on solid-body electrics. Instead, the difference in sound is just a result of electrical component tolerances. Potentiometers and capacitors can vary by 20% at the “same” value, and this can cause guitars to sound noticeably brighter or darker. Luckily, if your guitar sounds a bit brighter than Thom’s you can simply turn down the tone knob.
Thom with one of his vintage 1960s SG guitars at Lollapalooza Chicago on July 29, 2016 (lesserwiener).