Is any animal other than humans able to communicate with others in order to differentiate between various fruits, other animals, trees, places, etc.? How about using proper nouns?

Many animals are able to communicate with other members of their species in a very sophitiscate way that is able to address specific meanings such as differentiate different foods, dangers and yes, nouns.
I want to state that I believe in the theories of Frans de Waal (Frans de Waal) that address that every complex trait in humans being is the resoult of a evolutive history and hence it can be found, though in different state of complexitiy and development, in other species as well. And communication is one of these traits.

As I'm afraid that my question is going to be long, because I love this subject, I upload the video that talks about parrots and names here, if you want to find out more, is in the end of this answer :).
 
 

Here come some examples, some of them are more accepted than others.

Vervet monkeys make specifical sounds to alert of specific dangers, we could call them specific words.

 During the 1960s, researcher Tom Struhsaker of the New York Zoological Society carried out a study of vervet monkeys in Kenya's Amboseli National Park. He discovered that vervets have different alarm calls for their three main enemies: leopards, eagles, and snakes. Most interesting of all, he found that each type of call caused the vervets that heard it to behave in a very different way – a way that would help protect them from that particular predator.  When the vervets heard a "leopard" alarm call, they would scamper up the nearest trees. An "eagle" call caused the monkeys to look up at the sky and head for thick, low-lying bushes. Finally, a "snake" call made the animals        stand up on their hind legs and peer into the long grass around them.
Souce of the image and the quote: 2. The Secret Code of Monkeys. In spite that this investigation was very controversial in its time it is well accepted and often cited in studies and articles about animal communication.
 TalkBank Ethology Data: Field Recordings of Vervet Monkey Calls
(https://catalog.ldc.upenn.edu/LD...)

Dolphins using phones!.

• Bottlenose dolphins – It was long believed that dolphins were never able to demonstrate the  ability to communicate in their own language, but in light of a recent  discovery, this may have been incorrect. With the use of a CymaScope[24] researchers have discovered that dolphins are now using their whistles  to communicate more than just a simple hello to one another. They are  discussing their demographics:  names, ages, locations, genders, etc. It only makes sense that one of  the world’s most intelligent animals has a language that they use to  convey information to one another. Just like each person has his or her  own unique voice, each dolphin has its own unique whistle sound,  allowing each dolphin to maintain a separate identity, similar to  humans. Although this may be nothing spectacular, dolphins are able to  create new sounds and whistles when trying to attract a mate. Another  interesting point is that in most (if not all) groups, there is a  designated leader. The leader is responsible for communicating with  other groups’ leaders, discussing possible things such as demographics  or locations of food and danger. Remarkably, dolphins can hear one  another up to 6 miles apart underwater through a method called echolocation.[25]

National Geographic has an article outlining the successes of a mother dolphin  communicating with her baby using a telephone. Researchers noted that it  appeared that both dolphins knew who they were speaking with and what  they were speaking about. Nowadays, scientists are intrigued by the  dolphins’ language and are attempting to “crack the code.” Not only do  dolphins communicate via nonverbal cues, but they also seem to chatter  and respond to other dolphin’s vocalizations.[26]

Source: Animal language.
(On a side note, there are many more examples of animal communication in this article of Wikipedia).
Although, as it is easy of understand, this statement of dolphins using phones can be very controversial, the core of this research is to show that the signals used by dolphins are quite more complex and specifical than general calls or group songs as it was formely believed.

Bees dances to indicate where is the food. 

Curiosly this well acepted concept is being under debate in the last years. I put this here to put a specific example of animals talking about food, because this aspect of animal communication is not well researched, and also because if these bees don't dance for displaying where is the food, they are also showing a very complicated behaviour that deserves attention:

• Bee dance - used to communicate direction and distance of food source in many species of bees

And finally, my favourite one: parrots giving name to their chicks!

Scientists have known for some time that parrots use these  signature calls to refer to each other. Observing the process in captive  birds led researchers to wonder how wild parrots dealt with naming,  because it could show how names are given. The researchers felt there  were two possibilities for how parrots get their names: it could be  biologically innate (each bird names itself) or assigned by another  older bird, which turned out to be the case.
For the study, the researchers placed video cameras in 16  green-rumped parrot nests in Venezula. These birds are part of a large  wild population that has been living in nesting tubes rigged up by  scientists in 1987. The researchers then moved around the parrot eggs so  that half of the colony were raising babies that weren’t theirs  genetically. Recordings of the calls made by the parents before the  chicks were able to chirp , and of the calls once the chicks were  individually vocal showed that parents started making the calls when the  birds were very young. Additionally, the recordings showed that the  parent’s calls provided a basis on which the baby would imitate and  tweak their own name. The names bore more similarity to the parents that  raised the offspring, than the biological parents, suggesting that the  calls are in fact learned by the chicks rather than innate.
Parrots are not the only animals known to have names. In  addition to humans, dolphins also use specific names for each individual.

Source of the quote: Study: Parrot Parents Name Their Babies