Is that how much it would draw? I'd have thought there would be a 13A fuse somewhere limiting the current draw to ~3k RMS?

Maybe not.
If you plugged it directly into the mains then there would be no fuse.
If they used a cut up kettle lead with a 13a fuse then yes there would be a 13a fuse, but the fuse would not limit the current to 3Kw, the fuse would just blow when the power goes above 3Kw.
Now even if no fuse was used I think most house sockets use 20amp MCB's, but I've heard of some being 40amp (cooker feeds, etc.), I'm not a qualified electrician tho so don't take this as gospel.
The claim isn't far fetched tho, I've heard that some 80's PA cabs could be plugged into to the mains, but the reason they took the power was because the speaker (and enclosure) produced a high impedance (50 or 100Ohms maybe) around mains frequency (50Hz). Even at 40 amps, you can "only" pull around 10kw, so 13Kw would trip the MCB regardless...
I'll assume 50 ohms and 230v:
P = Vsquared / R
230 x 230 / 50 = 1058W so a lot of power but nothing crazy. Bloody loud tho I bet!
So if a 13 amp fuse was used and the sub still gave out sound after a few seconds, then it must have been pulling less than 3kw, meaning there is quite a high impedance at 50Hz... Still impressive tho, I imagine it must have a 6" voicecoil at least to deal with that amount of power...
What's better the top end JL subs or RE?
I'm not clued up on car audio, I'm more into PA.

The JL specs look a little more believable but they are stupidly expensive... If it were my money I'd go for a PD 1850 driver, one of the only drivers to have thermal compression given on the spec sheet.
I did put a horn in my car with 2 x 8" drivers tho, they get around 30W rms each and this makes things shake as it's a high efficiency system...