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You have to look at the meaning of "state". In a federal country, which the UK isn't, it means any of the separate parts that can partly govern themselves. Like the 50 states in the USA. Or the 16 in Germany.
But the word has other meanings. In the UK, it is used to mean the nation and its government. So "the state" prosecutes you if you end up in court for something criminal. The case is actually called "R v You" where R = Regina = the Queen, but in this context we're thinking of her as head of the nation/government/state. And we have state schools, the ones the government provides for free.
So similarly, there is a state pension, the one the government gives you according to law. Yep, "the state" is quite a difficult concept, but think of it as the government if that makes it easier. It's a bit more than that but that will do as a thought. It mostly gets there on thinking about what "the state" is.
Just for fun, my Mum has just had a letter saying her state pension will go up by 25p a week. This is the age addition because she reaches her 80th birthday next week. We had a good laugh over Sunday lunch on what she can actually do with that extra 25p. What actually IS there you can buy a whole one of with 25p? We couldn't think of anything.
The thing is that it was put into the state pension nearly 50 years ago when prices were so much lower. When I was a boy in the 1970s, 25p would be a reasonable amount of weekly pocket money. I could get some sweets with it, or even a paperback book to read. But no government has ever put the amount up since then, so it stays at 25p, even though now that feels like almost nothing.