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Trivium found on Wikipedia:

The guy that commercialised it was a teetotaller and wanted it to be called Root Tea, but because his target market were miners in Pennsylvania, he opted to call it Root Beer instead.

From my understanding, that title would be more accurate too, as it is produced from molasses with extract rather than grain mash, but my source is "skimmed Wikipedia" on both topics, so you should probably default to skepticism.

Either way, it apparently doesn't taste like beer, comes in both alcoholic and non-alcoholic* variants, usually doesn't contain caffeine and has a ton of flavours and variants from all over the world. If you care, you probably can find some.

*The process does involve fermentation, so I assume it will contain some ethanol still, even if it's below the threshold for the "non-alcoholic" label, in case that's an issue for you.