It's been noted in the past that animal names and varieties of plants are the most difficult to translate, and so we will be fortunate if we can determine a conclusive answer to these sorts of questions. I was originally thrown in understanding this question due to the KJV's rendering of 'asses', which I automatically assumed were cross-bred animals - but having read more on the subject I see that this was just a common word the translators used instead of donkey!
Can we confirm they're both donkeys?
As others picked up in the comments, it would appear there simply isn't gender neutral language for these animals, whatever they may be. In Hebrew we appear to have two separate words, which mandatorily specify the gender... but what is that supposed to tell us? Are these definitely two animals of the same species?
Since both words - the NKJV's "male donkey" (khamor / וַחֲמֹרִ֔ים) and "female donkey" (athon / אָתוֹן) - are used frequently throughout scripture, it would seem less likely that they would refer to anything other than donkeys. But let's see if we can discern anything from their usage in other passages. There are lots we can cross-reference, but I find Zechariah 9:9 proves the usage:
Rejoice greatly, O daughter of Zion!
Shout, O daughter of Jerusalem!
Behold, your King is coming to you;
He is just and having salvation,
Lowly and riding on a donkey [khamor: male],
A colt [ayir], the foal of a donkey [athon: female].
So yes, they're defined as the two sires of a colt, and there's no compelling reason to suggest they're anything other than male and female donkeys.
If they're both donkeys, why separate them?
At first, my suspicion was that the author may have been following a standard list convention for animals - and so I compared this with the following passage:
Then Abimelech took sheep [tson], oxen [bakar], and male and female servants, and gave them to Abraham; and he restored Sarah his wife to him. (Gen 20:14 NKJV)
No donkeys present in the group, but we've got our pattern of "sheep, oxen/herds... male, female", so it could begin to look like a norm of speech, until...
Thus the man became exceedingly prosperous, and had large flocks [tson], female and male servants, and camels [gamal] and donkeys [khamor]. (Gen 30:43 NKJV)
...two hundred female goats [iz] and twenty male goats [tayish], two hundred ewes [rachel] and twenty rams [ayil], thirty milk/[nursing] camels [gamor] with their colts, forty cows [para] and ten bulls [para], twenty female donkeys [athon] and ten foals [ayir]. (Gen 32:14-15 NKJV)
Finally let's recall your passage:
He treated Abram well for her sake. He had sheep [tson], oxen [bakar], male donkeys [khamor], male and female servants, female donkeys [athon], and camels [gamal]. (Gen 12:16 NKJV)
As above, in each instance we can see that there are gender-neutral normal terms for 'flocks and herds' - 'tson and bakar', and camels [gamal] are also used neutrally. But donkeys have distinct words that separate them, whether for physical or practical or simply grammatical reasons... I doubt we'll ever know.
Is there significance in 12:16's placement/pattern?
Looking at the spread of different lists of animals in Genesis, it's likely that gender is an influencing factor in the lists, to a smaller or larger degree. This arrangement may have been preferred by the author or previous oral tradition as it keeps the 'male donkeys' with the 'male servants' and the 'female donkeys' with the 'female servants'.
However, Genesis 30:43 and 32:14-15 would appear to emphasise the largest or most valuable group of 'wealth' first, and this is exactly how we would talk about things in English - working from the largest or most valuable items to the smallest or least valuable. Towards an intuitive reading consistent with the other animal lists, I would suggest that Abram's belongings in 12:16 are largely being sorted primarily by their value/prominence, and (possibly) secondarily by their gender.
However, these are simply theories. Going by the four passages in front of us, there is no obvious norm of how these things are listed, and so my theory is trying to make the most sense of the common factors we see between them all.