I still don't understand it...
Keysight has an official sales office in Germany, somewhere near Stuttgart. They have official Distributors in Germany, for example DataTec. So why on earth should a German winner have to pay a single cent to receive his prize? Winning a prize in a lottery is tax free here. There are no customs duties to be paid if the prize is delivered from a German distributor to a German citizen. And there is no VAT to be paid. All prices on goods, targeted at private individuals, have to be given including the VAT. The winners scope is free, it has no price. So 19% VAT from zero (the price the winner has to pay for the scope) ist zero, niente, null. Keysight simply can't charge VAT from the winner (private individual) for a free scope. At least that's my understanding
You are right. If Keysight give you a scope that is already legally in the EU, you owe the taxman, at least in the UK, nothing. It is
Keysight or perhaps their distributor who has to account for VAT. Keysight may be much more used to dealing with an EU business than a consumer, as the latter would probably buy from a distributor. If you are a business, you would know what to do with an invoice marked with zero charge for a business gift plus a VAT charge at fair value[1], although such a taxed gift would be a remarkably odd.
If you are consumer, such an invoice would be a mystery. And in many EU countries it would be unlawful to give a consumer such a VAT invoice. Consumers are supposed to be charged the whole price, including VAT. So if Keysight thought it was fair to give a gift but also charge the recipient the VAT they would really have to offer the scope at about a sixth of its price rather than as a gift, and take the VAT out of that.[2]
[1] Keysight do not have to pay VAT on the full list price if they don't want to. The usual selling price would be a better estimate of true value, and result in somewhat less VAT!
[2] The alternative is to 'give' the scope to the recipient at the US factory gate and leave the recipient to pay the import charges. But then they wouldn't be able to involve a distributor.