Can I ride a small moped/motorcycle elsewhere in Europe, with a UK driving license with "p" category entitlement?

I think @JoErNanO is largely correct that the p category is “solely valid in the UK”. It came automatically (no additional testing) with a car licence up until 2001:

If you have a full car licence which was obtained before the 1st February 2001 then you automatically have a full moped licence. This means you can ride a moped without any prior training, nor will you have to display L-plates plus you are able to take passengers.

AM effectively replaced p in 2013:

The law on Motorcycle testing and licensing changed (yet again) on 19 January 2013, introducing new classes of vehicles, tests and licences. It's all about harmonising with Europe, so that we can trust Jonny foreigner's licence and they can trust ours.

This last source also mentions:

a dash of Civil Service incompetence and , to the best of our understanding, you now get …

New licence holders under 17 need to be careful they are actually licenced for the Moped they are riding. We doubt Plod will even care but an Insurance Company looking for a way out?

You honestly couldn't make this up

Strangely there's no differ with Moped

Full car licences issued before 1st Feb 2001 automatically gave you a full moped Group P licence (how mad)

New licences (Group AM) are now a recognised European category so those will have to take the AM test and the CBT will be time limited again - it's a mess!

So, regarding age limits, even in the UK there was a difference between p and AM of some theoretical significance. However, the difference between the classes is 5 km/h in designed top speed. p has the higher allowed limit (50 km/h rather than 45 km/h, both cylinder capacity not exceeding 50 cc).

As mentioned in a comment, I don’t know (can’t quote) that a UK licence allowing 50 km/h is ‘legal’ where up to 45 km/h is the requirement, but it does seem likely (subject to a point later).

The short comments quoted above were to indicate that the UK law has been rather complex this century and that logic may not enter into the matter. Even harmonised Europe is still not consistent as shown by the age variations for AM mentioned in one of the links in the OP.

But it the above of much relevance? I think not because a visit to Vesping Barcelona as mentioned in Where can I rent a 125cc scooter in Barcelona by the month? the requirement is:

Vespa 50cc. European Driving License: AM, A, A1, A2, B. International Driving License: A, B

For the latter I think an IDP (International Driving Permit) is meant – available for £5.50 from “a Post Office near you” where reference is made to The AA where for Spain:

All valid UK driving licences should be accepted. Acceptance of driving licences that are not of the European Communities model cannot be guaranteed therefore drivers may wish to voluntarily update them before travelling abroad, if time permits. Application for D1 (in Northern Ireland DL1) is available from most Post Offices. Alternatively, older licences may be accompanied by an IDP.

However, in the list of countries for which an IDP is required, Belgium, France, Germany, Switzerland, The Netherlands and goodness knows how many other European countries do not feature. (Italy and Portugal are the same as Spain.) But maybe The AA has overlooked mopeds or oversimplified the requirements.

The terms from RivieraScooter.com for 50cc are:

Must be over the age of 18. For people born on/after 1/01/1988, the BSR (Scooter Training Certificate) or Drivers Class B is required.

Mention there is made of “Drivers License information” eg

125cc & MP3: Must be over 20 years of age. Full Motorcycle License (Permis A) or 125cc License (Permis A1) required.

(where Permis A is “Full Motorcycle License”, and Permis B [presumably the Drivers Class B mentioned] “125cc Motorcycle License”) but not of AM.

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In a comment to Is a driving licence needed to drive a scooter in Ibiza? I quoted from the English version of a website for a company that hires scooters in Spain:

You need to come with your passport or identity card, a credit card to pay a security deposit and your driving licence. To rent a scooter with Cooltra you must be in possession of a valid driving license and valid under the laws of the related country.

Here "related" looked to me to be "not Spain" but the Castillian version of the corresponding FAQ has a somewhat different:

Deberías presentar el pasaporte o el documento de identidad nacional, una tarjeta de crédito para realizar el depósito de seguridad y el permiso de conducir. La ley de España permite conducir scooters de 50cc y 125cc con el permiso de conducir de coche. Necesitas un permiso de conducir válido para España (vehículo o moto) por ejemplo, el permiso de conducir estadounidense solo debe ser traducido por un traductor oficial.

The last sentence I translate as:

You need a licence (car or motorbike) valid for Spain for example, a US driver’s licence must be accompanied by an official translation.

I doubt USA has a p category for driving licence's and am inclined to believe what is good enough from USA is likely to be good enough from UK, ie an IDP, which is the recognised official translation. p or AM then being irrelevant where a full car licence is held.