That is the case for aircraft operated by e.g. KLM. Some aircraft are named after birds, some after cities. Is that common for other airlines as well or something that only KLM does?
The practice of naming aircraft probably comes from the days of the flying boat, and the naval tradition of naming ships - the Pan Am Clippers had many named aircraft, and Pan Am kept up the tradition into the Jet Age. Other airlines of the period had similar naming practices (I'm not sure who started it, but Pan Am's fleet was well known for it).
As a marketing tool telling folks they're flying on "Clipper Emerald Isle from New York to Dublin" may have been more attractive to passengers than "Pan Am Flight 12345" or N880PA. Particularly before the commoditization of air travel, when flying was still a new and novel form of transportation, having a connection back to familiar "sailing" metaphors may have helped get passengers on board and could help differentiate your service from another airline.
Modern airlines don't seem to push their aircraft naming as much as historic carriers did, but as SentryRaven noted there are a few airlines that still name their ships - in fact Lufthansa paints their plane names onto the fuselage, though I'm not sure if they announce them to passengers.
A skilled marketing department could build a brand around ship names, but I'm not aware of any airline doing so today.
There are also other airlines which name their aircraft, e.g. Lufthansa.
To understand where this comes from, I recommend reading the blogs below, excerpts quoted here for a quick intro.
Lufthansa naming conventions as a sign of our time
The "Herborn" brings the total number of aircraft named after German cities and states to over 300. This naming convention has a long tradition at Lufthansa. The first Lufthansa aircraft was named in 1960. The idea was to express the company's solidarity with its German homeland – not just with the major hubs and cities but also with the regions where a large portion of Lufthansa passengers and employees come from.
KLM has always had a rather unique approach to naming its planes. It’s unclear who first came up with the idea of giving names to aircraft, but we do know that our first aircraft got its name in a completely different way than those in our current fleet.
KLM’s first Koolhoven actually had a nickname, “Dikke Dirk” (Fat Dirk), which was inspired by the last two letters of its registration, H-NADD, but also aptly described the unusually chubby fuselage of this aircraft type. “Dikke Dirk” was followed by “Piet Haas” (Peter Hare), because by then the Netherlands adopted the PH registration, which is how the Koolhoven registered PH-AES got the name “Piet Haas” (“haes” closely resembles “haas”, the Dutch word for “hare”). All this is much the same, I suppose, as people naming their car after the letters in its registration.
Let’s move on to the first “real names”. Not surprisingly, bird names were very popular. Vast flocks of bird names, in fact. Most KLM staff and aficionados are familiar with the “Uiver” (Stork), “Snip” (Snipe) and “Pelikaan” (Pelican), which all made legendary flights. The name “Uiver” certainly harks back to the era in which the plane was named, because most Dutch people nowadays would call a stork an “ooievaar”.
What do the “Borobudur”, “Museum Square” and “Blue Kestrel” have in common? They all crisscross the globe proudly displayed on KLM aircraft. But why do KLM aircraft have names? And who thinks them up?
KLM is one of the few airlines that gives its aircraft names. And I have to admit that I was unable to find out why. What I do know is that our very first aircraft, a Koolhoven FK-33 acquired in 1925, was known as the “Dikke Dirk” (Fat Dirk). The aircraft did have an unusually broad waistline, but the name had more to do with the last letters of its registration: DD.
What are KLM’s aircraft named after?
Airbus A330s: City Squares Worldwide
Boeing 737s: Birds
Boeing 777s: World Heritage Sites
Boeing 747s: World Cities
Boeing 747 Freighters: Ships of the Dutch East India Company (VOC)
Boeing 787-9 Dreamliners: Flowers
Southwest Airlines name some but not all of their aircraft; these are typically either limited runs for marketing (such as the Sea World, NBA and Sports Illustrated partnerships), aircraft dedicated to a particular state in Southwest's "home territory" (Arizona One, California One, Missouri One, New Mexico One, Lone Star One, etc), or else the first planes in some significant change to Southwest's fleet (Heart One and Heart Two were the first to be painted in the Heart livery that replaced the earlier Canyon Blue; the first Canyon Blue livery, in turn, was borne by Spirit One). A few are "signature planes" dedicated to notable Southwest employees or aviation figures in general, such as Fred J Jones. The full list of special liveries is on Wikipedia.