I think that the most straightforward answer to this question would be to suggest that, in Rabbinic Hebrew, קן ("nest") is also used metonymically to refer to the bird itself. So, for example, throughout Tractate Kinnim of the Mishna, where kinnim (קנים, "nests") is a way of referring to the birds.
There are numerous examples of rabbinic metonyms. Another would be "house" for "wife". In Shabbat 33b, to give but one example, Rabbi Shimon bar Yochai's wife is referred to as his "house". Jastrow lists this meaning as the fifth definition of בית. He also lists "birds" as a definition for קן together with its primary definition, the nest that they are in.