Hm, yes, so there is at least one degree program in "Climate Sciences" (they use the plural, not the singular, but that's a minor point). So I should amend that particular claim to: most of the people working in the field called "climate science" do not have degrees in "climate science(s)". Most of them seem to have degrees in physics, meteorology, "earth science" or some variant thereof, or geology.
I don't think this materially affects the rest of what I said.
http://www.swissuniversity.ch/unibe/climate-sciences/phd-pro...
"The PhD thesis usually consists of three to four peer-reviewed scientific articles"