Do the majority of Muslims accept that they're only Muslim because their parents were?

Even though it's true most would never admit it , I find it amazing how they are dying to get to countries with constitutionally protected rights of choice and as soon as they get there they impose on themselves the same oppression they left. It's similar to former prisoners who find it difficult to cope with freedom.

I guess.

As per Islam, if an individual is worthy of being Muslim because of their decision to believe, it would not matter if they were born to a Muslim family or not, Allah aza wa jal is the one who guides people so he would have guided them, had he so willed. The may have started off as kafirs due to their parents, but once Allah aza wa jal sends guidance, it's the same decision to make, believe or disbelieve not because your parents say so, it is your own decision always has been always will be.

there are many people who belong to Muslim families who aren't Muslim and vice versa. Me for example.

To be honest, that's the same concept with every religion. If you were born into a Muslim family, you would typically be a Muslim, if you are born into a Christian family, you would be typically born a Christian, if you were born in a atheist family you would be typically born an atheist. That being said, While you grow and mature you also start learning about your religion and about many other religions as well. That's why a lot of people convert into a different religion. I mean; here's a short story to explain the situation.

Billy was born a Jew into a religious Jewish family just outside of Oklahoma City. While learning about his culture throughout his childhood, Billy felt Judaism just wasn't for him. He then decided to convert to Christianity which made more sense to Billy in his heart.

Hope I helped.

One seems to believe that the faiths of the parents predispose all of a faith when many will chose other faiths or not to believe at all. King of a lame opinion.