https://youtu.be/jVhlJNJopOQ?si=GYj6XGNv2AMAZqCg
But, the reality is 99.9999% of websites can use one of the free Google fonts. The other .0001% could too, but they want to show that they are not like all the other websites.
So, yes, people would notice if most brands started "feeling" the same.
I do not believe this for a second. Has there been any serious research done on this?
Typography is visual design after all, and is not any different from other design elements we see all around us. It's obvious that it has an impact in how we perceive and experience brands, and the source of any text we read.
[1]: https://www.monotype.com/company/news/monotype-study-shows-t...
A sibling comment pointed out font psychology, which you might want to look into. Most of this is rooted in visual design, which is an inexact science, so you probably won't find the type of studies you want to see, for the same reason you won't find studies about other art forms.