Whilst there's an implication of strength through unity, it's not exactly what the Roman symbol means., which is a symbol of the office. Whilst it's suggested by the symbol, to make the leap from Fascism to Fasces is overly simplistic.
Fascism draws it's ideological roots from the Fin de siècle zeitgeist of the late 19th Century. This period was distinguished by popular perceptions of civilisational decline, social degeneracy and decadence, and pessimism.
In terms of ideology
The major political theme of the era was that of revolt against
materialism, rationalism, positivism, bourgeois society, and liberal
democracy.[5] The fin-de-siècle generation supported emotionalism,
irrationalism, subjectivism, and vitalism,[6] while the mindset of the
age saw civilization as being in a crisis that required a massive and
total solution.[5]
Different scholars have offered some definitions as described on Wikipedia
Roger Griffin describes fascism as "a genus of political ideology
whose mythic core in its various permutations is a palingenetic form
of populist ultranationalism".[29] Griffin describes the ideology as
having three core components: "(i) the rebirth myth, (ii) populist
ultra-nationalism and (iii) the myth of decadence".[30]
Robert Paxton says that fascism is "a form of political behavior
marked by obsessive preoccupation with community decline, humiliation,
or victimhood and by compensatory cults of unity, energy, and purity
The historian Zeev Sternhell has traced the ideological roots of
fascism back to the 1880s, and in particular to the fin de siècle
theme of that time.The fin-de-siècle
mindset saw civilization as being in a crisis that required a massive
and total solution.[68]
Overall we can see the same themes being repeated, the myth of a degenerate society, civilisational decline, and the need for national "rebirth", often through armed conflict.
In regards to it being synomousous with the Nazis, I think the term has become a perjorative, rather than holding any specific ideological meaning. It was always better applied to the Italian Fascists (who coined the term). Some quotes demonstrate:
George Orwell wrote in 1944 that "the word 'Fascism' is almost
entirely meaningless ... almost any English person would accept
'bully' as a synonym for 'Fascist'".[59]
Professor Richard Griffiths of the University of Wales[64] wrote in
2005 that "fascism" is the "most misused, and over-used word, of our
times"
The same can be applied to Nazism. Especially in the wake of the recent presidential election, it has become a popular term on the Left to describe any and all they deem to be a Trump supporter or a member of the political Other (for example: 4chan: The Skeleton Key to the Rise of Trump).
In present day usuage, "fascist" (or even Nazi) no longer really implies ideology. As Orwell, the former can be used to imply anyone with controlling or bullying tendencies, and the latter, can merely imply controlling tendencies see: The Soup Nazi
