How can I create a similar intro to the one in the video below using Final Cut Pro X or Motion and possibly a plugin? Or is it only possible with After Effects?
You could easily create that in FCPX, no plug-in necessary. The various circles can just be still images that you apply rotation and scaling to. (I'd make them large and then scale them down rather than making them small and scaling up.) I'm not sure if there's a title that exactly matches what they do, but it wouldn't be too hard to do that sort of titling. Worst case, you can pick up Motion for < $50 and use the text sequencer to do it.
You can try to mimic that by importing your alpha keyed assets and apply scaling and rotation but you won't be able to get that motion blur without a software like After Effects or Motion.