Are Book ISBNs in the United States a Monopoly?
Yes.
Am I right in perceiving Bowker as a government-sanctioned monopoly in the United States market for ISBN numbers by effectively eliminating competition?
You have presented no facts that are tied to government action. As far as I can tell, Bowker is "operating under a grant from the International ISBN Agency" and "has the exclusive right to assign ISBN prefixes and ISBNs to publishers located in the United States." But if you have additional facts, you should put them in the question.
I guess if you want to consider the lack of anti-trust actions against Bowker "government-sanctioning," then you can call it that, but that's not a legal opinion.
You say the government has "effectively eliminat[ed] competition." But you also present no facts that establish that allegation.
In most countries, the International ISBN Agency has entered into an agreement with a single entity to make them the exclusive local ISBN provider.
Is not $125 for a single ISBN rather exorbitant, considering that a single book often requires multiple ISBNs if published say, as a paperback, hardcover, and e-book---not to mention subsequent editions?
Whether this is exorbitant is a matter of opinion. But if one is publishing multiple editions of a paperback, hardcover, and e-book, it may make more sense to buy 10 ISBNs for $295 rather than to buy them individually for $125 each.