The word was invented by Richard Dawkins, an evolutionary biologist. He described ideas as reproducing with variation under selection pressure, which is the definition of the evolutionary mechanism.

There is a more recent use of ‘meme’ to refer to what we used to call an ‘internet image macro’. So the meme of the idea ‘meme’ has itself evolved to have a more specific variant of the general idea.

I think I see how you're thinking about it, to see how I see it, consider the following: Memes don't have to be inherited from genetic biological parents. Memes can have their own memetic parents. Like the joke "I feel like I'm the average of my 5 best friends" - our habits, quirks, idiosyncracies, all fit into the framework. It just needs some reproductive medium (human brains). That's why evolutionary biologists would be interested in studying this and things like it. There are epigenetic traits that exist and aren't genes and might be able to self reproduce. To keep the analogy going, consider that things that are catchy are easier to have this "memetic reproduction" so a catchy song might get sung a lot, serving as its fitness condition. So it's a generalization of biological genes to not just dna and rna and physical molecules but any substrate that can carry information. It's familiar to how we call it a 'viral' video.

Memes are inherited. You brush your teeth because your parents do. Memes are subject to selection via biological fitness. The people who didn't brush their teeth died sooner of tooth disease, so those who brushed their teeth had more children. Memes mutate. You don't brush your teeth identically to your parents.