The story is egregiously wrong.
In terms of track gauge (the distance between the two rails), the most common gauge nowadays is standard gauge, 4'8½" or 1435mm. This gauge was chosen by George Stephenson, who developed the first practical locomotive--and having developed the first practical locomotive, he was a strong influence on a lot of early railroad engineering choices. But standard gauge was far from standard--in the early 1800s, every railway chose its own gauge and gauge incompatibility was the norm. As the railway networks grew and interconnection was more important, there was a push to unifying on gauges, and even large networks of different gauges switched in the names of efficiency (the southern US, then on 5' gauge, switched over 11,000 mi on track on May 31, 1886).
As for why 4'8½" was favored over other gauges... well, it wasn't. Stephenson appears to have relied on compatibility with wagonways of the coal mines he worked with. But there was equally no standardization of widths there as there was with early railroad gauges--you end up with track gauges that range about 4'-6'. And from the operations of railroads, it does seem that optimal track gauge is around 5' anyways. So the real reason for 4'8½" gauge isn't "that's the width of two horse butts", it's "that's a reasonable gauge for putting a wagon on rails."
But, to be honest, none of that discussion actually matters. You see, what matters for the SSRBs is not the track gauge, it's the loading gauge--that's what determines the dimensions of stuff you stick on cars. Loading gauge is far less standardized than standard gauge--whereas the US, Europe, and the UK all share the same standard gauge, they have wildly different loading gauges. Whereas the US standardized on a 10'8" loading gauge, Europe generally runs on 10'4", and UK on a paltry 9'6".
Finally, if you look at the dimensions of the space shuttle's booster rockets, they are ~12'2" in diameter. Recall that the standard US loading gauge is 10'8"--a foot and a half shorter than their diameter. So clearly the booster isn't even based on the standard-shape-of-a-railroad car, although that doesn't preclude it from being based on the actual practicable loading gauge of any route.
Still, the last line ("The tunnel is slightly wider than the railroad track, and the railroad track, as you now know, is about as wide as two horses' behinds.") is complete and utter hogwash. The tunnel (I'm guessing Moffat Tunnel?) is probably about 13' wide, which is... a far cry from "slightly wider" than the railroad track at 4'8½".