Unfortunately the answer is almost certainly simply "No, absolutely not".
It's just a payment for a service - exactly like when you pay wait staff a "tip" at a restaurant.
The relationship Twitch viewer <-> Twitch performer is absolutely identical to BTS <-> BTS Concert Goer.
Obviously, when you buy concert tickets to see the boys, that's not a "gift". And if you, say, leave 10 bucks in the jar of a piano player at a hotel, that's also not a "gift" - it's just a normal payment. Exactly as a tip for wait staff is a normal payment.
Further confusion:
Note that a "gift" to a charity is totally unrelated here. (That only applies to actual charities, ie orphanages, homes for dogs with three legs, etc.) (Those are "501c3" organizations in US tax law.)
Even more confusion:
There's another way in tax law that the word "gift" is used - "business gifts". Say you're a company such as TireDiscounters. You give away pens with the logo on the pens. Another example is you're a freelance programmer and at the end of each year you send each client a bottle of scotch. {As a curiosity, these days there's a limit of basically 25 bucks to such "business gifts".} I just noticed online some twitch viewers were asking "Can I claim my tips as a 'business gift'" ... the answer is just No, it's utterly unconnected.
The logical error?
OP, notice you say ...
"They receive nothing in exchange for the contribution beyond a brief "holy crap, thank you so much!" ..."
That's completely wrong, again, exactly like the BTS performance or a waitresses' efforts. It couldn't be clearer that the Twitch celeb is a performer, and that the viewer is a viewer - it's totally cut and dried.
Again - confusion here may be with charitable causes. Sometimes you get a story about Little Johnnie who's dog lost a leg. The story makes it to TV. And some zany and kind Texas Oil Millionaire sends Johnnie a thousand bucks to buy a new leg. That's a charitable act, and Johnnie is not in any way a performer - he was just going about his business when he made national news. There is absolutely no relationship between that scenario and a Twitch performer. (And furthermore if I'm not mistaken, in that actual example the Oil dude can not claim it as a charitable gift - that's precisely why in such situations where donations are needed, someone sets up a 501c3 organization you see? Again Johnnie and the missing leg are just wholly, totally, completely, absolutely unrelated to BTS, waitresses, or Twitch performers/viewers. To repeat, you can not just randomly give someone money and go "oh that was charity" .. the IRS have "thought of that one" :) But once again the Twitch performers/viewer situation is anyway just totally unrelated to a charitable gift .. a Twitch perform is as certainly a performer as BTS or a waitress, and the viewer is as certainly a customer as an audience or diner.)
The summary
It's completely identical to paying a waiter a tip - that's it.
The fact that the word "gift" is sometimes used rather than "tip" is totally irrelevant and meaningless. The two ways "gift" are used in tax law are not even vaguely related to the issue at hand.