I don't believe any situation can exist between individuals where an amount would be rightfully considered a taxable gift from the giver and taxable income to the recipient. People may argue about whether or not an item is a gift, but I've never seen any portion of tax code that indicates something can be both a gift and not a gift at the same time.
In US tax law double taxation is generally avoided. Counting a payment as a gift subject to gift tax on the side of the giver and also counting it as income subject to income and/or self-employment tax on the side of the recipient would not be consistent with that principle. A main component of gift-tax is that it is paid by the giver and the gift is tax-free to the recipient, which necessitates an either-or classification.
We've addressed situations where people try to claim a tip is a gift on this question: Reporting monetary gifts as a waiter/waitress? The conclusion is that calling it a gift doesn't change the nature of the tip, a service was performed and money was given as a reward for that service.
Twitch "donations" are typically no different, it's not something for nothing it's pay what you want for entertainment. There are multiple tax guides for streamers that indicate all of their income is just that, income. You posted a guide indicating the same. Gift-tax is paid by the giver and is tax-free to the recipient, the fact that streamers income is taxed as income demonstratesjust that their, income does not qualify as. By virtue of it being income to the streamer it cannot also be a taxable gift to the giver.
I said typically because a twitch donation could qualify as a business gift, but business gifts don't trigger gift tax. If one big streamer goes around to smaller streamers/channels and gives donation/tip, those could be viewed as business gifts (but most likely considered promotion/advertisement). A business gift is just an expense to the giving business, so it's not really the same as gifts in the context you asked about. Business gifts are subject to a set of rules about how much can be deducted and whether or not they count as income to the recipients.