Jun and other knowledgeable people on this forum will be able to confirm or dispute it, but my impression is that kata
simply refers to pre-decided series of movements - such as the basic aikido techniques, when done in an orderly fashion, whether or not together with a partner.
Kata can absolutely be done in pairs, like the kendo kata that are always done with a partner. Same in judo. So, basic aikido training is kata training.
Please correct me if I'm wrong
Anyway, there are lots of things that one can practice on one's own, for example any one of the aikido techniques. I am not sure about the fruitfulness of it, though.
Maybe the best solo practice is aiki-taiso stuff, like irimitenkan taisabaki, shomenate in shiho, funakogi undo, et cetera. Not to mention suburi with bokken or jo.