Rav Hirsch explains that the animals were shown to be the ones that Noachides were going to be offering on the altar to Hashem. I would consider this to be based on the Bnai Yisrael becoming a ממלכת כוהנים. Thus, the restrictions on Bnai Yisrael as compared to the rest of the world are analogous to the restrictions on the kohanim with respect to the rest of Bnai Yisrael. This analogy would also apply to the Shabbat. Kedushah (holiness) is the separation and restriction required of the kohanim. Just as the kohanim are separated from the remainder of Bnai Yisrael, so too are the Bnai Yisrael separated from the rest of mankind.
The point is that the animals with respect to Bnai Noach are such that the animals, when they were offered on the altar, could not be eaten. The Bnai Noach were allowed to eat any animal that had not been brought on the altar. Only certain animals were allowed on the altar, though they could be eaten if they had not been brought. Bnai Yisrael on receiving the Torah at Sinai, were raised in kedushah (holiness), so that they were only allowed to eat animals that were permitted to be brought on the altar. Animals that were actually brought on the altar were restricted even further.
Animals that Bnai Yisrael bring on the altar are restricted even more than other animals so that certain offerings may be eaten only by kohanim, and certain others only under specific circumstances by those who brought them (Korbon Pesach and other Shelamim as examples).
Note that Hashem did restrict the eating of flesh to an extent so as to show that it is only by the permission of Hashem Noach 9:3-4. Rav Hirsch explains this as well.
Rav Hirsch explains in [Noach 7:2}(http://www.chabad.org/library/bible_cdo/aid/8170#v=2&showrashi=true)
Of all the clean animals you shall take for yourself seven pairs, a
male and its mate, and of the animals that are not clean, two, a male
and its mate.
Although at that time animal flesh was not an object of human
consumption and the difference between "pure" and "impure" animals was
only made later by the laws given on Sinai, nevertheless already here
Hashem designates animals to Noah by the characteristic of being
"pure". So that there must, already then, have been some occasion for
making this distinction, and this occasion was offerings, for
Noachides also were not allowed - and are not allowed - to bring
other than pure animals as sacrifieces. (Zevachim 115a). So that it is
only that which is fit for all men to offer, that Jews are allowed
to eat. The Jewish table and the Noachian altar stand at the same
level. The choice of animals for both must accordingly have the same