I'm not an expert in radar, so this is a hobbyist's opinion, powered by being a boating/marine enthusiasts and ham radio guy. The ideas I'm tossing out here are probably worth what you paid for them ($0)
But, here goes anyway ...
The earlier automotive range finding chip radars (probably the only thing affordable) - had small parabolic fixed antennas because they needed to check the distance only in a certain direction. For a radar to "activate" a sensor on a tag, its beam would need to be steered. The sensor might report the ID when struck (although detecting the radar signal would not be trivial), and the radar could then maybe record the distance and position of the tag. The steerable antenna could be a parabolic antenna or a digitally steered antenna array, with the former being a mechanical nightmare when considering both horizontal and vertical planes, and the latter not uncomplicated with a half dozen antenna elements or more. In the case of the parabolic, the beam-width could be as small as 1 degree which might give good enough resolution, while the antenna array would give less resolution (5 degrees maybe?) - probably not good enough to activate the sensors of a single bird. All in all, the whole thing is probably a totally, totally, ridiculous thing to contemplate.
Like I said, I'd not seen any good answers to this kind of question, and your requirements (birds) make it even more difficult because they can work the x, y, and z axis coordinates, unlike for example a cow. And they're tiny. Great mind exercise to think about the problem though.
RFID has the same problems as the radar IMNEO (in my non expert opinion), in addition to not having any range capability. Technically, some tags have position via phase, but what I've read (below) says reflections may make the position unreliable:
https://www.quora.com/Can-the-position- ... r-3D-space
I don't know if that happens in many cases, so don't really know if it's an option. Maybe that article is similar to what you were looking at? For RFID and cows, one might plant tags in the pasture, and put readers on the cows. But then the dung could cover the tag, rendering it useless. This post may be dung.
Here is a bluetooth based position/tagging system, but it looks like only 2D, not the 3D you need: