Genesis chapter 2 presents a more detailed version of the creation of the animals and humanity that the terse presentation in chapter 1.
Regarding the creation of Adam, Genesis 2:7 (NIV) states
Then the LORD God formed a man from the dust of the ground and breathed into his nostrils the breath of life, and the man became a living being.
Regarding the creation of the animals, Genesis 2:19 (NIV) states
Now the LORD God had formed out of the ground all the wild animals and all the birds in the sky. He brought them to the man to see what he would name them; and whatever the man called each living creature, that was its name.
The Hebrew text (MT) for Genesis 2:7 and Genesis 2:19 use the same exact phrase for both the "living being" of verse 7 and the "living creature" of verse 19, נפש חיה, nefesh hayyah. Verse 7 indicates that the breath of God transformed the dust into Adam, the man, a living being, while verse 19 implies that the animals are also living beings, without stating by what means they became living beings.
So, already after verse 19 we have both the man and the animals called "living beings", nefesh hayyah, with no apparent distinction between them other than the fact that it is the man who gives the animals their names.
In the story of the creation of the woman in Genesis 2:21-24, there is no mention at all that she is a "living being", a nefesh hayyah, only that the man gives her a (generic) name, "woman"!
However, the next use of nefesh hahayyah, in Genesis 9:9-10 (NIV)
"I now establish my covenant with you and with your descendants after you
and with every living creature that was with you--the birds, the livestock and all the wild animals, all those that came out of the ark with you--every living creature on earth.
makes it clear that the women with Noah and his sons are also nefesh hayyah, living souls.
It appears that there is a progressive contraction of the narrative in Genesis 2 - the breathing of life in Genesis 2:7 is omitted from the creation of the animals in Genesis 2:19, and both the breathing of life and the fact that the woman is a living being are omitted from the creation of the woman in Genesis 2:21-24. But as we see from subsequent verses, we can't conclude from these omissions that there was no divine breath of life or that the woman was not a living being.
So, to answer your question, the narratives of Genesis 2 provide no basis for concluding that men, woman or even animals were created differently as living beings, nefesh hayyah, except that the woman was created from the man rather than from dust. Note however, that many other verses such as Genesis 1:28 and Genesis 3:16 do establish an order of authority among men, women and the animals.