Firefox takes privacy so seriously that they fail to detect you are on Firefox, if you are using Firefox!

I looked because I was curious, and on both desktop and mobile, in all browsers, the site's topnav includes their Mozilla logo, a Download Firefox button, and links to a couple of other Mozilla sites about Internet Health and Donate.

I imagine they could still choose to hide the blue button when you're on Firefox, but that wouldn't save you any vertical space, since the topnav menu of links and logo would remain.

It's there always in the same place so you can download Firefox for someone else, or for another device.