As a European, the phrase 'tail number' was unfamiliar so I did a few searches.
It seems to have derived ( in the USA ) from original airline fleet-numbering practice, where the aircraft would have an arbitrary and unhelpful state- or even city-issued registration and a separate self-assigned fleet number painted on the tail, which was easier to handle for scheduling purposes.
Once national N-numbering for registrations was introduced this became less common because the FAA permitted reservation of blocks of sequential registrations, but some large airlines continue to use a parallel fleet numbering scheme. And the phrase 'tail number' stuck in parlance instead of 'registration'.
In Europe it is common to use to the last two or three letters of registrations repeated on the tail or nose-wheel doors as the fleet airframe identifier and 'tail number' is unknown.
That's a strange flight pattern, any insight into it? Erratically gain altitude, make a long run and then a tight spiral back down before returning to land?
There's a pretty strong weather system in that area, and that looks like a flight pattern to try to figure out what sort of risk the storm has in generating tornadoes. Head out in case there's another area that could use a little extra info, and then come back home.
