...50 amp cable so the whole house will be supplied during power outage...
The problem is 10 awg cable. 10 awg supposed to support 30 Amp.
Don't confuse the NEC's "Small Conductor Rule" (SCR), with actual ampacities for wires and cabling. The SCR only applies to site wiring, not appliances, which are rated individually by their manufacturer.
As
@Camofurever1 showed in that nice chart (first one, not the second), the current-carrying capacity of a wire is dependent on its insulation type and configuration, such as the number of current-carrying conductors in a bundle and the temperature ratings of the terminals at either end of the wire. It is the manufacturer's responsibility to figure this out for their wire type and configuration, and the UL or CE approval (although often self-declared) is your assurance that this was done.
Bottom line, you don't have the information required to make an absolute determination on your own, but you can at least see if the rating is in the realm of reasonability by comparison to tables such as the one posted by
@Camofurever1. In this case, it appears it is not, as even if high-temperature insulation was used, the terminals at either end (your mating cables) are likely not rated above 90C.
Sent txt to the seller and his answer:
"The short distance (1.3 ft) means that the resistance is low and the cable is resistant to high temp., so it can pass 50 A current".
I find it hard to believe. Any opinions? Are there other similar gadgets? Don't want to overload the generator.
The reality is that yes, a short conductor will allow higher current for the same temperature rise, under the assumption that either end is connected to a large conductor. In fact, I do this every day in my business, cooling smaller conductors by connection at either end to larger conductors... but what I'm building is installed in military test sites and not your home. I've never seen anyone ever operate a mains voltage circuit for consumer appliances on wiring beyond its standard temperature current rating, on the bet that it will be cooled by connected devices... it's just not commonly done. If legal, I don't know, but it's terrible practice.
I would personally not use this device on a 50A breaker, but you haven't stated what size breaker you're operating this circuit on.
I'm an MSEE, so I can give you all the theory in the world, and reasoning behind various conventions and the NEC. But I don't have the NEC memorized, like any good commercial electrician should, so their conclusions would overrule mine. I work on very high power industrial and military electronics, not residential or consumer products.