At the level you are talking about a football club is, strictly speaking, a company, either private or public. It is in the business of selling tickets, sporting paraphenalia, advertising in stadiums and on jerseys and all the other ways that sports teams make money. The company does that by producing a successful sporting team. Successful team means that people will pay to watch them, and to buy crap with the team name. Advertisers then follow.
The club isn’t the team, but it’s very closely tied to it, both financially and in the public’s mind. So when someone refers to “Manchester United Football Club”, they may be referring to the company or the team. You just have to divine which from context.
Football clubs started out as just plain clubs, like pretty much any other club. Think about any of the clubs you were in at school, or the local Rotary or Lions, but on larger scale. There was a president, secretary, treasurer etc and, of course, and members. Most football clubs are still like that: small affairs in small towns with maybe a couple of dozen members. At that level most of the club members will be players, or their famies. The point of the club is to play football. The club organises matches against other teams. It organises the team membership at various grades or appoints coaches and captains to do so. It raises money for upkeep of the training fields and clubhouse and to purchase equipment and uniforms. All the things needed so that a couple of dozen men can play football for a couple of hours on the weekend.
Some of the clubs associated with the more successful teams grew so much, had such high membership and made so much money that it became necessary to transform from clubs into companies. But they kept the name of “Football Club” because that is what the public knew. But they are no longer actual clubs in any sense of the word.
The actual supporters are usually still members of clubs, but now that is membership of the club supporter’s club, affiliated with the company-called-a-club. So, for example, Manchester United Football Club is a company, and has no actual members. It sponsors Manchester United Supporters Club, which normal people can join. But that club doesn’t have anything to do with playing football. It’s effectively just a fan club.
I’m not sure if that clarified it or made it worse.