Personally I'd jam it, here's why.
I have TPTK against a completely unknown, his line is so weird I want to find out what kind of garbage he does this with and add a note for future encounters (provided he calls, which he will).
The amount of hands that beats us is less than half of the hands he can play like this. Worst case scenario, he does this with a set, 34s or A9. He can also be doing it with a boatload of flushdraws, 9x, any pocket pair, hell, maybe even some gutshots. Usually you'll run into some kinda A-high flushdraw, I can easily see him holding the Ah8h, which could (somewhat) explain the way he's playing this.
If that was me I would have taken the free card on the turn and paid off any bet on the river that wasn't a huge overbet. Get the info as cheap as possible. The downside is losing value in this given situation (IF we are ahead), but the information is worth alot more in the long run.
By the way, the call on the flop is fine, provided you think hitting an A or K will give you the lead, just be aware that Ah or Kh could screw you over real bad. I'd only make the call if I "knew" villain wasn't drawing.