The Chrome distribution deal with Adobe was always Windows-only, as far as I know. So I'm not surprised you're not seeing it on Linux. That said, Adobe is just showing whatever they were paid to show, for whoever the target audience is. With Firefox on Windows as of today it shows checkboxes for some antivirus programs instead of Chrome; I assume McAffee outbid Google here.

With IE11, which does support ActiveX, it shows ... a Chrome download checkbox. And the Google toolbar.