I have the same bass, except mine is a 2007 one. I guess that's not a factor to take into account, BTW, I've always seen it as a pretty consistent series. If others have different experiences, please chime in.
I suspect the obvious wrong info can possibly be due to ignorance rather than malice: typos like "EMC" are often indicators. That said, I don't really know the quality of such pickup and I'd take into account the possibility that it's poor.
It'd be great if they also add the stock one, which they might not do due to ignorance (ie: they were sold the bass with that pickup and "bought" the tale) or because they don't want to contradict themselves if that was a lie in the first place.
However, the stock pickup is the same spotted on Am St. P basses from around 2008 to (IIRC) 2012. Alnico rods, plastic bobbins. IMO, it does the job and is nothing special. JimmyM used to say he hated it.
My tactics would be: I'd provide good evidence (at least two sources, photos...) to certify it's not a stock bass, and would do so in a calm way, without blame. Then I'd use it as a reason to bargain a bit. If the deal's good, I wouldn't worry that much about the factory pickup, you can probably pick up a SPB1 somewhere for less than you saved and end up with a better pickup anyway. I used to have a SPB1 in mine for a while, still have it in my drawer.
Or you can go with a Geezer's EMG, or higher end stuff... I just mentioned the SPB1 as it's a "classic" with good value for money, if that's your priority.
That's of course the way I see it, YMMV. We're not talking about a pickup gutted from a Custom Shop, or even a "59" from Mike Dirnt's sig model, which seems to be worthwhile and not sold aftermarket. This IMO is not the case, even though I DO LOVE the series.
Actually, I once did a short comparison where you can A/B a few lines played with the stock one and the SPB1. Keep in mind these are only guidelines, not final benchmarks to spot all subtleties involved:
