...no? Technically it's given name-family name as in all "Western" societies I'm aware of; in practice for pretty much all formal/official purposes family name comes first and is the more important one since that's what just about all records are indexed by, which AFAIK is the norm about everywhere anyway. And in less formal/bureaucratic use it's still polite to address strangers by the family name (again hardly unusual) so people often introduce themselves in reversed name order, and everyone can tell which are the given and family names anyway.
And anyone who gets a stick up their butt about it will be viewed as an OCD twat.
Kind of like if someone introduced themselves as "Smith, John" to you I daresay you would have no problems whatsoever deciphering the meaning...
As an aside it isn't exactly an unusual practice for Western publishers to simply include a primer page which explains the name order and suffixes for the benefit of newcomers to imported Japanese manga. And while that doesn't quite work for anime it REALLY shouldn't be too hard for the audience to figure out the basics to the degree they know what's going on; I see no virtue in losing nuances solely for the sake of dumbing everything down to the level of the absolute lowest common (Anglo-American) denominator who can't wrap his head around the idea of some things being done a bit differently in other corners of the world.
And for the sake of comparison it's not like say US media gets localized for differences in education system organization when exported elsewhere, ergo I have to try to remember which order the whole sophomore-freshman-senior-junior thing was in again ('round here we just refer to high-schoolers and equivalents by which of the 3 years they're in)...