I've never 'got'

I never got why middle class is a thing

Divide and conquer.

I never got why middle class is a thing. There's working class and the capitalist class, who own the means of production.

Well, that's essentially how Marx saw it but a major problem with that pov is that even back then only a very small percentage actually owned those means of production. And since then industrialisation, automation and computerisation has reduced the percentage who are actually involved in the production bit.

The only difference between working class and what people suppose middle class means, is having higher wages i guess?

Bit more than that. Education, expectations from life, view of personal position in society, ways of expressing these things... one reason the middle classes expanded so much post WWII is that a lot of people from working class backgrounds got access to education to a level none of their predecessors had. And discovered they were now talking a different language and viewed as different by the social groups they came from. Some of them wrote plays about it. Involving kitchen sinks, for some reason.

How you talk about things has looong been a class identifier in this country at least.

Is middle class the accomplice class of the capitalist class?

Is a teacher a capitalist lackey training children to be slaves or a dangerous subversive teaching them unpatriotic and subversive beliefs? For some time now, the Tories have mostly taken the latter viewpoint and also applied it to a lot of other bits of the middle class (all those lefty lawyers, you know).