Sherlock sits like this because Sherlock is clever

This method of sitting and style of leather armchair is in the public domain and thus no iteration of Sherlock Holmes can claim it as its sole intellectual property. They only didn't use it in Mr Holmes because Ian McKellen is like ninety and he couldn't manage it without his knees cracking.

(NB no, Google Images doesn't have any pictures of Robert Downey Jr's Holmes sitting like this. My point still stands.)