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Good question. Initially, at the beginning of the war, these vehicles were covered by the Portuguese Insurance Association:

As far as I know, iIt seems there are now Ukrainian green cards for these vehicles:

Portuguese insurance cannot, as a rule, insure a car with a foreign licence plate. To do so, the vehicle would have to be legalised and registered in Portugal.

I know it's UK and not Portugal but there does seem to be some exemptions - from UK Gov website -

Exemption from vehicle tax and registration

If you have been issued a visa under one of the Ukraine visa schemes, you do not need to tax or register your Ukrainian plated vehicle in the UK.

This applies for up to the first 3 years (36 months) from your entry into the UK, if you were:

    issued a visa under any of the Family, Homes for Ukraine Sponsorship Scheme (HFU) or Extension Ukraine visa schemes

    initially granted an 18 month stay under the HFU and were granted a further 18 month visa extension under the Ukraine Permission Extension (UPE) scheme

This exemption will apply for up to 4 years and 6 months (54 months) from the date you entered the UK if you were initially granted a 36 month stay under the Family, HFU or Extension Ukraine visa schemes and have been granted a further 18 month visa extension under the UPE visa scheme.

Eligibility criteria

You will only be eligible for the extended exemption for up to 4 years and 6 months (54 months) if you applied for UPE before the expiry of your current Ukrainian scheme leave.

@Strontium

In Portugal there are also tax exemptions for these cars (eg. IUC - imposto único de circulação). The question here is insurance, what happens in case of an accident.

Portuguese insurance cannot, as a rule, insure a car with a foreign licence plate. To do so, the vehicle would have to be legalised and registered in Portugal. - @JohnnyPT

I thought that this is the case. In case of an accident, I wouldn't want to deal with an insurance in Ukraine (if there is even one). I assume (!) that with a green card insurance issues are somewhat under control. Without, who knows. Thanks JohnnyPT for investigating this. Always appreciated.

I still see them all over Austria

@SimCityAT

As I am there (Vienna) at the moment I can confirm this. However, it seems that there is an agreement (?) that Ukrainian vehicle owners don't have the comply with the usual regulations. E.g. they don't get traffic infringement (e.g. parking) tickets. All fine with me, but I am worried about an accident involving one of these vehicles.

Considering the UK deal, owner of UA vehicles seem to get a very good deal.