To be honest, I don't understand all of the legal mumbo jumbo contained in these quotes, but at least it should put the debate on wildlife ownership to rest. I pinched these quotes from the Siskiyou County Farm Bureau site. There's a lot of information there, but I think these excerpts are faithful to the underlying arguement.
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In Baldwin v. Montana Fish and Game Comm'n., 436 U.S. 371 (1978,) Justice Blackmun explains the reasoning behind early claims:

"Many of the early cases embrace the concept that the States had complete ownership over wildlife within their boundaries, and, as well, the power to preserve this bounty for their citizens alone. It was enough to say "that in regulating the use of the common property of the citizens of [a] state, the legislature is [not] bound to extend to the citizens of all the other states the same advantages as are secured to their own citizens." Corfield v. Coryell, 6 F. Cas. 546, 552 (No. 3,230) (CC ED Pa. 1825).

It appears to have been generally accepted that although the States were obligated to treat all those within their territory equally in most respects, they were not obliged to share those things they held in trust for their own people. (emphasis added)

Geer v. Connecticut, 161 U.S. 519 (1896) In Geer, a case dealing with Connecticut's authority to limit the disposition of game birds taken within its boundaries, the Court roundly rejected the contention "that a State cannot allow its own people the enjoyment of the benefits of the property belonging to them in common, without at the same time permitting the citizens of other States to participate in that which they do not own." 161 U.S., at 530. (emphasis added)

"In more recent years, however, the Court has recognized that the States' interest in regulating and controlling those things they claim to "own," including wildlife, is by no means absolute. States may not compel the confinement of the benefits of their resources, even their wildlife, to their own people whenever such hoarding and confinement impedes interstate commerce. Foster-Fountain Packing Co. v. Haydel, 278 U.S. 1 (1928); Pennsylvania v. West Virginia, 262 U.S. 553 (1923); West v. Kansas Natural Gas Co., 221 U.S. 229 (1911). (I imagine the USO case will fall into this category)

And a State's interest in its wildlife and other resources must yield when, without reason, it interferes with a nonresident's right to pursue a livelihood in a State other than his own, a right that is protected by the Privileges and Immunities Clause. Toomer v. Witsell, 334 U.S. 385 (1948). See Takahashi v. Fish & Game Comm'n, 334 U.S. 410 (1948).

Yet, in his concurring opinion in Baldwin, Chief Justice Burger still reveal an attachment to the belief in common public ownership of game found within a State and a State obligation to manage as a public trust.

"The doctrine that a State "owns" the wildlife within its borders as trustee for its citizens, see Geer v. Connecticut, 161 U.S. 519 (1896), is admittedly a legal anachronism of sorts. See Douglas v. Seacoast Products, Inc., 431 U.S. 265, 284 (1977). A State does not "own" wild birds and animals in the same way that it may own other natural resources such as land, oil, or timber. But, as noted in the Court's opinion, ante, at 386, and contrary to the implications of the dissent, the doctrine is not completely obsolete. It manifests the State's special interest in regulating and preserving wildlife for the benefit of its citizens. See Douglas v. Seacoast Products, Inc., supra, at 284, 287. Whether we describe this interest as proprietary or otherwise is not significant."