I live in Sweden but I've never met anyone with this name. In fact there is only 88 women named this in Sweden. I haven't heard of anyone named Kaja who is from Scandinavia.
This name has been quite popular in Norway, although its popularity is declining.
-- Anonymous User 3/5/2012
One reason why this name is so rare in Sweden is probably the fact that Kaja is the Swedish name for the Western Jackdaw (Corvus monedula). There is also a derogatory term "fyllkaja" for someone who is very/often drunk; it literally means "drunk jackdaw" and may be derived from seeing jackdaws behave like drunks after eating molding berries.