Why wouldn't it be true? You saw it for yourself.
So anyway, I signed up to try and delete my account from there permanently. Apparently I can't do it myself so I asked them to do so. The "boss" of the website replied saying "It's public data, anyone can be on there. You'll be removed shortly.". So yeah. Is this true? Can any website add our PSN Profile with the updated data (trophies, about me, etc) without our permission?
Yes. The trophy servers are public and the trophy sites know how to pull data from them. Anyone can retrieve anyone else's trophy list through those sites just by putting in a PSN name.