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3 Answers
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Your discussing the cog sci called graphology. It doesn't really work in general cases but its a nice idea. I would say its a happy accident that they look alike. My letter R look like my father's R because I wanted to please him at that age so I mimicked his handwriting.

Graphology is the study of handwriting. As a theory or practice for inferring a person's character, disposition, and attitudes from their handwriting, graphology is generally considered pseudoscience.

Graphology has been controversial for more than a century. Although supporters point to the anecdotal evidence of thousands of positive testimonials as a reason to use it for personality evaluation, most empirical studies fail to show the validity claimed by its supporters.

Thing is, when we look at something for a while, it influences us to, in some ways, follow it. If I am not mistaken, a similar principle works for yawning and how a person can yawn just by looking at someone else who does. I wish to elucidate with a personal experience. In my high school, there was this girl with amazing handwriting. I got down with the pox so I used her notes to study. And soon, I realized I have started writing a bit like her. My letter were slightly tilted (hers was at a precise 80 degrees and was consistent across the page) and more cursive. My work became more neat like hers and far more presentable. And I wasn't making a conscious effort to mimic her handwriting at all. It just happens.

I don't know of any particular research that's looked at this, but based on general principles, I'd conjecture:

If a mother teaches their child to write, then it makes sense that they could share a handwriting style.

Within a society, there may be shared educational practices about how handwriting is taught, which could lead to similar writing styles. That said, such practices appear to change over generations.

I imagine that writing style is culturally specific. That said, some traits that are heritable might be relevant to writing style. For example, various abilities often have a genetic component. In particular, various forms of fine motor control might be relevant to writing ability and style.

Why any two people have similar handwriting could just be a coincidence.