I've only seen these names spelt as
Kaj and
Kaja in Sweden.
Kaj is a unisex name in Sweden. I think the female
Kaj is a short form of
Kaja or a nn for
Karin. Another nn for
Karin is
Kajsa, which is also used as formal name.
The Finnish form is
Kaija (short for
Katariina). There is also a Swedish word "kaja", which means the bird jackdaw. But this has nothing to do with the name
Kaja.
"You sought a flower and found a fruit. You sought a spring and found a sea. You sought a woman and found a soul. You are disappointed."
"It does not become me to make myself smaller than I am." (
Edith Södergran 1891-1923)