“ Out of the ground the Lord God formed every beast of the field and every bird of the air, and brought them to Adam to see what he would call them. And whatever Adam called each living creature, that was its name. So Adam gave names to all cattle, to the birds of the air, and to every beast of the field.” (2:19, 20)
In chapter one, verses 21, 24 and 25, the fact is repeated that God was creating and making creatures, in the water, on land and in the air, “according to its kind.” This is the word for genus and indicates limitations of variation. For example, canis, Latin for dog, including wolves and all varieties of dogs, domesticated and otherwise. Although Adam didn't speak Latin, the concept is that he just named the genus, not every currently existing species of dog. It is important to note that in 2:20, it says that Adam named “all cattle, to the birds of the air, and to every beast of the field.” Adam did not name insects or sea creatures. That would likely have taken a little more time than one day.
Bos is the genus name for cattle, that is, wild and domestic cows and oxen. A word commonly used in the Bible is cattle, or kine, without there ever being a distinction between a dairy cow, a meat cow or an ox (an animal which isn't actually referred to until the time of Jacob in Genesis 32). The same naming arrangement could have applied for animals such as horses, snakes, birds, etc. Again, Adam did not speak Latin or Hebrew, so genus names he gave to animals way back when may easily have become obsolete.
Excluding insects, which are not technically animals anyway, and other animals whose categories are not specified, it would seem feasible to allow for the naming of 1,000 or so “kinds” of animals and birds. When all is said and done, it was likely an amusing, typical day's work in the Garden of Eden.