As noted in the question comments, the kanji 円 was originally 圓. The nutshell version of the article Yang Muye linked is that monks developed a shorthand version of 圓 that looked like a box with a vertical line through it: . Over time, the shape of the surrounding box changed, likely due to the same anatomical and mechanical processes that inform any change in handwriting. became , and then that became the modern simplified shinjitai character 円.
門, meanwhile, is a pictogram of a gate. You might run across alternative form 鬥, but otherwise, this character is much more straightforward, both in semantic development and graphical evolution.
Ultimately, the resemblance between 円 and 門 is purely accidental, and has much more to do with the constraints placed on kanji shapes and the kinds of strokes used in handwriting.