The header image at the homepage is another size than the header image at the other pages. I don’t use the profile picture on my size, so now the menu I would like to have them the same size, because the menu at the homepage is floating somewhere in the middle of the image.
Since you’re not using the features of the home page template, I would recommend setting the home page to either the default page template or full width page template. That may result in the menu displaying as it does on other pages.
However, the home page header is purposely designed to display differently from the other pages. So, if that does not work, a CSS solution may be necessary.