This seems like a reasonable combination to me; the subwoofer was only rated 30w RMS according to the spec sheet, which a TPA3116D2 should be capable of even if it's an 8 ohm speaker. However as noted the crossover will be different (and appears non-adjustable), there could have been built in eq, etc. Still, if it sounds good at low volumes and isn't making mechanical noises when you turn it up, then I believe there is hope yet for this speaker.
What power supply are you using? Even though it will power up and operate on e.g. 12v, it will have severely limited power (which could explain sounding good, but only at very low volume). You will want 19 to 24v with 3+ amps (more is great but don't pay extra for it, at least for this project). If it's an 8ohm speaker, a 24 volt power supply would be best (~32w, 1% distortion, 8ohm, PBTL), but 19v should still get you ~20w into 8ohms. If it's 4ohm or less, then 16-19v is arguably better than 24v, but it's not critical. If you're not sure, I would try an old 19v laptop power supply because they are a good compromise and available cheap or free!