The EU country flag carrier I used to fly for allowed for small (can't remember how it was defined) cats and dogs to fly as "pets in cabin" with their (presumably) owners. Cats were always in a pet carrier; occasionally so were dogs but we carried a fair few dogs who were simply on a lead. This was in European shorthaul.
Young Lady once bought her cat via Easy to Liverpool back in the early 00's, from I think Paris, when she arrived the sh*t hit the fan. Her (reasonably acceptable in my opinion) excuse was that she was used to doing this around Europe, and didn't think the UK would be the exception (strict quarantine rules) and they were quite happy to let her board with catty. Oh how she threw a tantrum. This was filmed I think and shown on BBC? as part of some Easy documentary series.