As of the 38th launch, they do not name boosters cores, but they started numbering the cores, and printing that number just around where the legs attach at the base of the booster core.
The cores have serial numbers, four digits, so the first reflown booster was 1021 used on CRS-8 then SES-10. The Bulgariasat booster set to be reflown was first flown as Iridium-1 and is serial number 1029. 1026 is the Thaicomm-8 mission booster and is being used as a Falcon Heavy side core.
The first reused Dragon (CRS-11 after flying CRS-4) had an ISS 'kill' logo painted on it. (Much as fighter jets will have logos of various kills marked under the cockpit).
As of this writing only one core has reflown, second due in 10 days, and no signs of "kill" markings from previous flights on cores, yet.
A neat infographic of launches trying to visually capture as much info as possible is nice to add in here as well.