A Note on Languages
The English we hear in My Little Pony: Friendship is Magic is not, of course, really English. The look of the ponies' alphabet bears this out. It is Equestrian, and it is only translated to English so that we may be able to understand what is said by Twilight Sparkle, her friends, and the other beings she encounters. The translation is complete and absolute, but it is a translation nonetheless.
J.R.R. Tolkien, whose work has greatly inspired my own, adopts a similar conceit in both The Hobbit and The Lord of the Rings. The English in those books is not English, either; it is Westron, also called the Common Tongue. It is rendered into English to ease our understanding of the story, and the rendering is so complete that even names are 'translated' from Westron into English, to a degree that some Westron names sound quite different than their English counterparts. For example, Meriadoc Brandybuck's 'real' name is Kalimac Brandagamba; Merry's name is changed in the text of The Lord of the Rings to better preserve the 'sense' of what his name would mean to speakers of Westron. I presume that Fluttershy, Pinkie Pie, and all the rest have names that similarly sound nothing like how we hear them, but what we do hear has been translated to preserve the same sense they have in the original Equestrian.
Perhaps even less noted in Tolkien's work was what I like to call 'relational translation.' That is, Westron was translated into English, and other languages were translated into other historical languages based on their relationship to Westron. The native language of the Rohirrim from The Lord of the Rings, for example, was translated into Anglo-Saxon, because the Rohirrim's language has the same relationship to Westron that Anglo-Saxon does to English. In the same way, the language of the Men of Dale from The Hobbit was translated into Old Norse, because it is related, and not related, to Westron in the same way that Old Norse is related, and not related, to English.
I do not have Tolkien's gift for languages; indeed, I have always been rather wretched at languages which are not my native one. Nevertheless, I have attempted to undertake a similar 'translation project' in both It's a Dangerous Business, Going Out Your Door and Besides the Will of Evil. The Equestrian that the ponies speak has been 'translated' into English. Meanwhile, the other languages in the story have been similarly 'translated' based on their relationship to Equestrian.
In the case of the deerfolk, it is important to remember, as Chapter 8 mentioned, that the six deer species were at one point distinct civilizations with their own languages. Based on each of those languages' eventual relations to Equestrian, I have 'translated' each species' language as follows:
Red Deer: Italian
Mule Deer: Spanish (particularly Castilian Spanish)
White-Tail Deer: Portuguese
As you can see, some of the translated languages are related to each other in the same way that some of the human languages are related to each other. I suspect that, at some point in the extremely distant past- perhaps tens or even dozens of thousands of years ago- the elk, red deer, mule deer, and white-tail deer all had a kind of common proto-civilization whose common language had the same relationship to their later languages that Latin does to the Romance Languages. Similarly, I suspect that the caribou and the moose, both species from the cold northern reaches, at some point mingled to an extent that they shared a common language that had the same function as the language which gave rise to the Germanic Languages. Regardless of primordial association, however, by the time of The Comrades the six species were distinct, with their own distinct languages- until the Elements of Harmony brought them together.
When the six species decided to unite and form the deerfolk, it was determined that they should have a new language that bound them all together, something wholly distinct from anything any of them were speaking at the time. It fell to the elk, always the greatest writers and poets among the deer species, to invent this language, which they did. This is what I have chosen to call Laewtil, which, if I remember my work from Dangerous Business properly, translates as 'antler tongue.' Laewtil too is translated, but because it is an invented language with vague similarity to the Romance Languages, I have chosen to translate it as another invented language with vague similarity to the Romance Languages: the Elvish that Tolkien himself dreamed up. Particularly, I have chosen to work mostly with Quenya, as that has the most complete grammar and vocabulary; however, where I have been unable to find suitable Quenya words to complete a sentence, I have reluctantly used Sindarin words.
I admit that this project has been only haltingly achieved; as I previously mentioned, I am not good with languages. Any help I might receive in the future on proper use of languages which are not English would be immensely appreciated.
One final note: you have noticed that the various deer species have different-sounding names. Though all the deer species adopted Laewtil, most of them chose to continue naming their children in their original languages, as a way of holding on to their history. Only the elk, the inventors of Laewtil, took it on completely- they fully displaced their old elkish language with Laewtil, even going so far as to name their children with names derived from Laewtil. That is why, while moose have German names, white-tails have Portuguese names, and so on, the elk have Quenya/Sindarin names.