Took me a while too, I think GP and the text about the designer considering where the 'logo' is anchored in designing the, err, 'logo' are talking about the full name as shown at https://kagi.com/assets at the top. I don't know where else that's used though.
It's definitely confusing to call it a logo in talking about designing the logo (let's say icon) though.
Mind you before seeing that about the 'g' I thought the dog was their 'logo'. I'm surprised the designer didn't insist on some simplification, a stronger, single identity; not that I know anything about design or marketing.