Pull quotes are short excerpts from the presented text. They are used to pull a text passage out of the reader’s flow and give it a more dominant position in the post or the article. The pulled out passage should be mentioned somewhere in the text as this pulled out passage refers to the current text, it's a kind of internal quote. Here highlighted blocks are independent blocks of text, not a quotations at all. Therefore it's inappropriate to use <aside> for ".box-note" blocks here.
Here is pretty simillar case: https://instantview.telegram.org/contest/informer.rs/template33/issue2/