In medieval Europe, money bags were often made of cat skins. Making items out of cat skins would be almost unthinkable today. When and how did our attitudes towards cats change?

The idea of 'animal cruelty' in Europe is largely a construct of the 19th century. Attitudes towards animals in general had started to shift during the 18th century, probably as a result of 'politeness' developing as a dominant mode of behaviour among the middle and upper classes. So in this period, many animal sports such as bull-baiting and dog-fighting were still common and legal, but were associated with the working classes and were criticised by many 'polite' social commentators and letter-writers as violent and cruel, often on Biblical grounds. In 1800, the British MP Sir William Pulteney attempted to pass a law banning bull-baiting, but it was narrowly defeated in Parliament. Similar failed attempts were made in 1809.

The anti-cruelty movement got under way properly in the 1820s. In 1822 a bill was passed through Parliament which outlawed the ill-treatment of cattle, and in 1824 the RSPCA (Royal Society for the Prevention of Cruelty to Animals) was founded in Britain. (The 'Royal' bit was added in 1840 after assent from Queen Victoria). The RSPCA popularised the anti-cruelty message through religious sermons, and cash prizes for the best essays written against animal cruelty.

So where does that leave cats? Cats had been domesticated in ancient Egypt, so the idea of the cat as a pet to be cherished was an old one, alongside the usefulness of cats as pest controllers. Cats were a feature of early-modern farms, villages and urban areas, and their popularity as pets in the 17th and 18th centuries is clear from the frequency with which they appear in portraits with their owners, especially high-status women. But they were also killed for fur, as you note, and subject to ritual killing due to their associations with the Devil, especially in 17th century France (see, for example 'The Great Cat Massacre' by Robert Darnton). Again, it's only in the 19th century that these practices begin to be universally regarded as cruel.

Another contributing factor to our changing attitude to cats was the development of cat fancy, or cat breeding. The National Cat Club was established in the UK in 1887, to promote and regulate the breeding of pedigree cats.

This answer brought to you by me, and my cat who is sitting next to me as I type.