Yes, Lithium Ion batteries are capable of delivering the power required to turn over an engine. The thing I am most experienced with is the kind of batteries you put in remote controlled devices, so I will use those for reference. For all intents and purposes they are basically a consumer-friendly version of the bare cells you will find in the devices you are talking about.
How much power (simplify this down to current, since current is a function of the power, voltage and resistance, and for all intents and purposes the voltage and resistance is relatively constant) a lithium ion battery can deliver is a function of how much surface area the electrodes within the battery have. How much voltage the battery has is a function of how many cells it has.
Because more surface area = effectively higher current capabilities, a larger battery pack generally means a pack that can deliver higher current. The batteries discharge rate is measured as a function of its capacity as a results of this. A battery that is 10000mAh (10Ah) and has a 15C constant discharge rating - C standing for the capacity of the pack - means that the battery can be discharged constantly at 15 x 10A = 150A. In other words, you can buy a battery pack capable of delivery 150A at 14.8v off the shelf today from hobbyking (I can't suggest hooking one of the directly up to you car's battery, though).
To add to this, the devices you are talking about aren't just a battery pack on it's own, they also include all sorts of charge circuitry and whatever else the manufacturer has put in them... as a result of this I would say it's safe to hazard a guess they aren't just battery packs like what I have described here (Even though we have come to the conclusion a battery pack on its own can deliver the power required for to turn over a reasonably sized modern engine), I would say they probably have a small capacitor bank alongside the battery. Basically, capacitors can deliver significantly higher sustained current, but do not have much capacity. Think of a capacitor as a bucket, the cable from the battery to the capacitor as a hose, and the battery as a 50 gallon drum. You can only fill up the bucket as fast as the water moves through the hose, but once the bucket is full you can dump the entire bucket at once on whatever it is you need. Because a starter only needs it's peak current for a very short time, you could use a capacitor bank as a buffer from the battery to the starter to turn over the motor with very high currents.
In short, a battery alone could turn over an engine relatively easily, but a battery alongside a bank of capacitors would do it without breaking a sweat. Throw in some circuitry for over-current protection and you can make the whole thing work perfectly with little worry of it going wrong and blowing up in your face.
With regards to asking for a similar battery online... its possible the manufacturer of the device is sourcing directly from a battery manufacturer to their desired specifications. This is a pretty heavy duty pack that I can't imagine has much mass-market appeal due to what would be incredibly rapid discharge times running at its rated load... You can find 2S 8000mAh 60C lipos online, put two of these parallel and you effectively have a 4S 8000mAh 120C lipo... this is 1S over the one in the video but you get the idea. You could maybe team up a 1S and a 2S but i'd be worried of getting cells that are too different.. a least with 2 identical pack's you have more chance of them being near identical.
It's also possible that the 120C rating is the pack 'burst' rating, not its continuous current rating. this is more likely as it's very easy to find 3S 60C 8000mAh packs with a burst rating of up to 120C, the manufacturer might be boasting the burst rating for this specific product as that is the bit that will matter with this, since the peak load with be short lived.
If you want to see how high continuous load you can go this route, there's also products like this 5600mAh, 90C pack, thats 504A continuous current. At that rate you should be able to deliver 504A for ~30 seconds before the pack is discharged. that's a lot better than the 350A for 10 seconds you are talking about in your question and again, this is just from hobbyist gear off the shelf. The manufacturer of these devices probably has a much wider range of suppliers to choose from, I can't tell if they are the producer of the battery pack in the device (since it's branded the same as the actual device) or if they are re-branding the packs, but it only strengthens the argument that these are packs produced specifically for this purpose.