Nothing to do with VAT. The place of supply for VAT purposes is the country where the journey starts. If you take a taxi from England to France, you pay VAT in the UK. On the return trip you pay VAT in France. The taxi operator must be registered for VAT in both England and France if it does that, regardless of where it is based for other purposes.
Having said that, I believe Über claim that your contract is directly with the driver, and as the driver almost certainly earns less than £82,000 per year, they don't have to register for VAT. Most minicab and taxi operators do this, it isn't unique to Über.
What happens is that the Dutch company pays a royalty to a company in Ireland. The Irish company pays a royalty to another Irish company that is tax-resident in an offshore tax haven such as Bermuda. That way you get the money into an offshore tax having while paying very little tax. If a UK company paid the royalty directly to a tax haven, they would have to deduct a withholding tax of 20% from the payment.