Yes, the terms are related. Dictionary.com says that fasciae comes from Latin for band, bandage; "akin to fasces". The origin of fascism is given as:
Origin: 1915–20; < Italian fascismo, equivalent to fasc(io) bundle, political group (see fasces) + -ismo -ism
Wikipedia explains the connection:
The term fascismo is derived from the Latin word fasces. The fasces, which consisted of a bundle of rods that were tied around an axe, was an ancient Roman symbol of the authority of the civic magistrate. They were carried by his lictors and could be used for corporal and capital punishment at his command. The word fascismo also relates to political organizations in Italy known as fasci, groups similar to guilds or syndicates.
The symbolism of the fasces suggested strength through unity: a single rod is easily broken, while the bundle is difficult to break.