But they're not a Fascist icon either, despite what your comment suggests. The fact that it appears on the Senate building probably refers to the Roman meaning. The point of TFA is pointing out that these symbols are NOT owned by the Fascists or Nazis but should instead be returned to their cultural origins.
The roman meaning is very much fascist, if phrased in modern terms. Fasces were carried by lictors accompanying a roman official, symbolizing the "imperium", or absolute, king-like power, held by that official over his domain. While for a time, the imperium of lower officials was somewhat curtailed during the republic, military commanders with imperium still held the absolute and immediate power over life and death of their subordinates (other officials still had power over life and death, there just was an opportunity for citizens for appeals and vetoes by equal-ranked officials) . Lictors with their fasces were not only bodyguards but also a small police force at the discretion of the commander, who would seize an offender, hold him for a drumhead trial by the commander and then punish (e.g. execute) him if so decided. Under the later roman dictatorships the dictator (imperator, emperor or caesar) of course also held those powers, with the same symbolism of imperium represented by the fasces. Fasces always have been a symbol of capital punishment, justice by drumhead or thumbs down, and absolute power.
No, the Roman fasces are not fascist. Imperial Roman power is not fascism, even if fascism tries to fashion itself to Imperial Rome. If A claims inspiration from B, it does not mean that they're the same thing nor interchangeable.
The difference between the two is 2000 years and one being an entire civilization that spanned hundreds of years, and the other a relatively short lived political ideology, based on reactionary nationalism, autarchy and later anti-semitism (racial purity was more a Hitler kinda thing).
But, more importantly, "imperium" doesn't mean what you think it means. There are two meanings to "imperium", one being the overall realm that Rome controlled, the Roman empire. This is the meaning you did use in your above straw-man. Usually, this meaning of imperium is represented by symbology such as the eagle and the S.P.Q.R. signature. Italian fascists also used those symbols, but the international fascist movement mostly didn't, because just the Italians wanted to rebuild their Roman empire. However, the fasces represent the second meaning of "imperium", which is the absolute power of one single individual official to rule and impose order in his assigned domain or subdomain. They represent the ordering principle of fascism and nazism that a strict hierarchy of individual leaders ("duce" in Italian, "Fuehrer" in direct German translation) should rule the state ("Fuehrerprinzip" in German). Fasces also represented the primacy of punishment and violence in imposing order, of swiftness and immediacy in carrying out justice, very much what modern day fascism wanted to return to. So Roman fasces are representing what modern day fascists intended for the principles of leadership, justice, punishment, and order to look like. But beyond those aspects, there can be no inference made, because modern-day Fascism encompasses more aspects than just those.
That there is a 2000 year difference between the cultures in which to interpret the aforementioned symbolisms is a problem indeed. However, it is a problem that was created by people like the americans thoughtlessly using symbols because "well, Rome was a republic, we want to build a republic, let's just steal all of their iconography". Even back in that day, the actual meaning of fasces was known and clear. It was known and clear to the Romans as well. Hell, look at the list of uses for fasces in the wikipedia article: mostly police forces and departments of corrections, monarchies and off-with-the-heads mobs (such as the early french post-revolution order).
You'll even find it in the currently used Seal of the United States Senate[1]!
It's also often seen displayed prominently on stone buildings across the US. I was recently walking around downtown St. Charles, IL and there were Fasces overhanging the entrances of an old stone apartment building.
[0] https://en.wikipedia.org/wiki/Fasces
[1] https://en.wikipedia.org/wiki/Seal_of_the_United_States_Sena...
edit: correction s/Nazi symbol/symbol Nazis used/ thanks to commenter.