Why are they not able to put tags/strings on the bird's feet?
How do they transport the birds around the Temple? Not in cages?
hgschild, Spring Valley
1) It seems that tying tags is not such a good solution. One can learn this from a Sugya in Beitzah 10b. The Gemara there discusses preparing birds before Yom Tov so that they should be ready to be slaughtered on Yom Tov. Since they must be specified before Yom Tov, one encounters problems if the birds became mixed around after Yom Tov entered. One could ask the same question in that Sugya: why were tags simply not placed on the legs of the chickens and then one would know on Yom Tov which was which?
2) It seems that the answer may be that the Gemara there (end of 10b) states that if one ties chicks together, "Menatchei Ahadadei," they become detached from each other. The Mishnah Berurah (OC 497:41) writes that it is normal that animals become detached from the way they are tied up.
(See also the Mishnah in Bava Kama, end of 17a, which discusses the "Dalil" tied to the legs of a chicken. Rashi (DH Dalil) writes that anything tied to the leg of a hen is called a Dalil. One sees from the Sugya there that the Dalil may often become detatched from the chicken.)
3) One of the kinds of birds offered as a Korban in the Beis ha'Mikdash was the Tor. This is translated by Onkelos (Vayikra 12:6) as "Shafnina." In Tehilim 84:4, we find that the "Dror" bird is also translated as "Shafnina." This suggests that "Tor" and "Dror" are synonymous. The Gemara (Beitzah 24a) tells us that the Dror cannot be tamed. Rashi there (DH b'Tzipor) writes that it knows how to dodge around the corners of the house and it cannot be caught. In addition, the Gemara states there that even other birds, which appear to be more tame than the Dror, come back to their cages only at night, and some do not even do that.
It is difficult to restrict some birds to the cage, so this would not always be a feasible solution in the Beis ha'Mikdash.
Kol Tuv,
Dovid Bloom
So how do these women transport all these pairs of birds at once once they buy them to the mizbayach?
How do smart woman know which bird is which?
Chaim
Certainly, whenever possible the birds were presumably transported to the Mizbe'ach in cages. My argument was that it is not always possible to keep birds in cages for longer periods of time.
Again, the wise ladies no doubt often used tags and strings to identify the birds, but sometimes they come lose, and the Mishnayos discuss scenarios where something went wrong.
Kol Tuv,
Dovid Bloom