If you think these animals say their own names, you are wrong
Anyone who's heard an English cuckoo knows where it got its name. Cuckoos make a two-note call, the first sounding like the vowel in "cook" and the second like the vowel in "coo." Between the two, there's even a little catch. This bird says its own name.
Or maybe not.
Aside from not knowing that that's its name, there's that matter of how the bird actually "says" it. Birds don't really use their tongues as we do. They don't make an actual "k" sound, or pretty much any other consonant sound either. They also don't move their tongues around to make vowel sounds as we do. And, obviously, they don't have lips.
So why do we hear cuckoos saying "cuckoo"? There are a couple of reasons.
First of all, birds have other ways to alter the shape of their resonating space. They can control the opening of the beak and the size of the space at the back of the throat. They also have a more complex voice box that can change a lot of the tonal details before they even go into the throat. You don't need a human mouth to make sounds that sound like speech, after all — the speakers on your stereo aren't shaped anything like your mouth.
Secondly, the sound cuckoos make isn't really exactly "cuckoo." It's really more like a "woo-'oo" than a "cook-coo" or "cuckoo." But humans tend to process animal sounds into the kinds of sounds we would make. And we like the crisper onset and the clearer break of the "k" sounds. So when we imitate the bird, we make an English-sounding word out of it, "cuckoo," and so we call the bird a cuckoo. And when we hear the bird, we hear it saying "cuckoo" because we're expecting that.
Let's look at some other birds that seem to say their names — plus two mammals that do as well.
You've probably heard this bird. It has two calls, one a three-note song and the other two quick high chirps followed by several lower, raspy notes. It's the latter one that is said to sound like "chickadeedeedeedee." But listen again: You won't hear a "ch" or a "k" in there anywhere, just a couple of sharp breaks in the notes. We humans are completely unable to imitate those quick breaks. The best we can do is to give an impression of the quickness and the high pitch with "chicka" — and then we approximate the repeated part with a "dee." When we hear the chickadee, we can hear that "chickadeedeedeedee" — but a human saying "chickadeedeedeedee" sounds exactly nothing like a chickadee. Still not sure exactly what it does sound like? Listen to it, and many other birds — including some of the other ones I mention here — at the Cornell Lab of Ornithology's site, allaboutbirds.org.
This New Zealand bird has a simpler, easier name call. It makes a sound that starts high and tight and then breaks to a lower, more open-sounding note. An acoustic phonetician would tell you that speech sounds are made and identified by the relative strength of sets of harmonics of the base note, and when a kea makes its noise, the first strong set of harmonics above the base note moves from low to high, while the next strong set above those moves from high to medium. But if you watch one of these parrot-like birds make the sound (as well you may if you visit New Zealand's South Island, because they're not exactly shy), you won't see it move its tongue like you would for "eeaa"; it modifies the sound by other means. You also won't see it say a "k" before the "eeaaa" — because it doesn't. Again, we just want to insert one there to emphasize the sharp beginning of the sound — that's how we would do it.
This bird makes a sound you'd probably hear more as "whoooop-whoop," though if you said "whoop-whoop" the way you would read it off the page, you wouldn't sound like this bird. The ancient Romans called it "upupa," and in English we used to call it "hoopoop." Somehow that came to be change to "hoopoe"; I don't know whether that was for reasons of decency or dignity, or simply random mutation, but I do know other versions of the name from earlier centuries include "whoophoo" and "whopee." Meanwhile, the hoopoe doesn't give a whoop whether it's getting its name right… it just gives a "whoooop-whoop."
The three-part song of this bird has gotten it the Portuguese name "bem-te-vi" and the Spanish name "bien-ti-veo." If you listen to it, you will probably hear something like a raspy "Ih! Ee weeo," largely due to the first note being high and short, and then next holding a little lower but dipping in the middle and then gliding downward. How does this make it "kiskadee"? Well, you insert "k" and "sk" to indicate the sharp start and almost as sharp end of the first part, and then stick in a "d" instead of a "w" probably because you're thinking of a chickadee…
One sound the curlew (or one kind of curlew) makes has a long liquid sound between two high chirping sounds, and it seems that a long time ago people in France found that "currrrrrrleu" was a good approximation of it. I think it sounds more like a squeeze toy or someone cleaning a mirror, but, then again, if someone were to try to spell either of those sounds, they might end up with "curlew" too. Because how do you spell a sound humans can't quite make?
If you know this bird's name, you can hear it when you hear its call. On the other hand, if someone told you it was called a "Quaker reel" you'd hear that. Or "potpourri." Or any of quite a few other possibles. What is clear is that there's a sound at the start that reminds us of an "o" or "u" or "w" sound (because the first and second sets of harmonics are both low); there's a catch between the first and second parts that we'd think of as a version of a stop consonant like "p" or "k"; the second note is longer and lower than the first and is mainly a trill; and the third note is much higher and quicker and trails off a little at the end. There just happen to be a lot of plausible sets of English sounds that match that. Picking one is sort of like seeing animals in clouds.
This bird's name comes from its call — or some part of its actually incredibly varied vocal repertoire — supposedly sounding like "Bob o' Lincoln." To my ears it sounds more like an excited R2D2. But the many little notes coming quickly one after another have a little resemblance to the sound we make it we say "Bob o' Lincoln" quickly — or "little kitty litter" or "abalone goblet" or… Well, someone picked this version, and it stuck. Meanwhile, the bobolink goes on babbling and doesn't care what it's called.
This cute little South American rodent spends a lot of its time burrowing, but when it makes a sound, the sound is something that sounds like "tuc-tuc-tuc." So why call it "tuco-tuco"? Because you find it in countries where they speak Spanish and Portuguese, and in either language "tuc" is not a well-formed word, but "tuco" is. And we got the name from them.
Here, at last, is an animal that we can actually think of as able to say its name as we do. It's a sloth, a furry slow-moving three-toed jungle creature — but a mammal with a vocal tract a bit more like ours, although of course not able to carry on a conversation. When a female ai calls out for a mate, or when a male or female is in danger, it makes a high-pitched sound like a woman screaming (not very laid-back for such a chillin' creature), and sure enough, the way its mouth starts wide open and then closes down makes the cry sound sort of like a human "ay" or "ai" — but a very high one!