When the mishna states the time for the beginning of saying Shema in the morning, it states that when you can tell the difference between blue and white. The gemora brings a breissa saying between a dog and a fox, a donkey and an "arud". Is this the same time? What is the significance of these examples? Why did the Mishna stick with blue and white? Does anyone speak on this?
On a P'shat level, the significance of these examples is that they are the natural ways to tell how far along in the day you are. Rebbi Meir wanted to relate his opinion that Keri'as Shema may be recited from such-and-such level of lightness in the early morning hours, and so he gave a concrete sign of when that time has arrived -- the time at which one can recognize the difference between a dog and a wolf.
The two opinions in the Mishnah (difference between blue and white, and difference between blue and green) are definitely arguing, because one (the difference between blue and white) is certainly earlier than the other.
The three opinions in the Beraisa (difference between a dog and wold, difference between a donkey and a wild ass, and the time at which one recognizes an acquaintance at a distance of four Amos) are definitely arguing, as is evidenced by the fact that Rav Huna and Abaye rule in accordance with one of them. If they were all the same time, then there would be no need to rule like one of them. The Tana'im are arguing about the exact definition of "the time of arising" (Z'man Kimah), which is the time that the Torah requires one to recite Shema in the morning.
However, are all the times mentioned in the Beraisa arguing with the two times that are mentioned in the Mishnah? (Logically, if one of the opinions in the Beraisa agrees with one of the opinions in the Mishnah, then obviously the other two times in the Beraisa are arguing.) There does not seem to be any indication whether any of the times in the Beraisa agree with any of the times in the Mishnah. However, it could be deduced that the time that is given in the Mishnah by the anonymous opinion and the time of the "Acherim" in the Beraisa are the same. Why? Because the Gemara rules that the Halachah is like "Acherim" (when one recognizes an acquaintance at a distance of four Amos). We know that the Halachah is usually in accordance with the anonymous opinions of the Mishnah (which, in our case, is the time when one can distinguish the difference between blue and white). If so, those two times are probably identical!
Indeed, the Mishnah Berurah (OC 58:2) says, "... Therefore, the Chachamim established its time to be from when one can distinguish between blue and white which is the same time that one recognizes his acquaintance from a distance of four Amos."
Going back to your question about why the Tana'im chose these particularly signs to tell us the time to say Shema, the P'NEI YEHOSHUA discusses their significance in a brilliant novel approach to the Sugya. The P'nei Yehoshua explains as follows:
The opinions given by the two Tana'im in our Mishnah are based on a Beraisa in Menachos (43b) which derives from the verse, "And you shall see it (the Tzitzis) and remember them (the Mitzvos)," that one sees this Mitzvah (Tzitzis) and remembers another Mitzvah, and what Mitzvah is that? The Mitzvah of Keri'as Shema.
It is clear from this Beraisa that the Mitzvah of Keri'as Shema depends upon the time that one is able to see the Tzitzis -- that is, the time that one can recognize the Techeles (the blue) in them. The Tana'im in our Mishnah are just arguing about how distinguishable the Techeles has to be (must it be distinguishable from green, or only from white). That explains the argument in our Mishnah.
The Tana'im in the Beraisa are arguing on another, related point. The Rabanan understood from the words "u'v'Lechticha ba'Derech" ("... and when you travel on the way") that the Torah is telling us that the time for the morning Keri'as Shema is from the time when one travels on the roads. When is the time that one travels on the roads?
Rebbi Yochanan states that one should only embark on a journey during daytime, lest he be attacked by wild animals which he cannot see and from which he cannot flee. When there is enough light in the day that one can see a wild animal approaching and flee in time to save himself, that is the time that one may embark on a journey, and, consequently, that is the time at which one may begin to recite the morning Shema.
This explains why Rebbi Meir says that the earliest time to recite Shema is when one can distinguish between a dog and a wolf , because at that time one may embark on a journey, because one is able to see and avoid any approaching wolves. Rebbi Akiva maintains that it must be light enough to recognize a wild ass, because only at that time would one embark on a journey. "Acherim" say that Shema may be recited from the time that one recognizes an acquaintance at a distance of four Amos, because at that time of morning one may embark on a journey without worrying about being attacked by armed marauders, and since that is the time that one would "travel on the way," that is the time at which one may recite Shema.
I hope this was helpful. All the best,