I don’t mean to pile on, but this issue does come up a lot:
Underlying this way of translating the text is an assumption that the writer had no way to clearly refer to a past event. Apparently Hebrew is so limited that both “formed” and “had formed” must be written the same way. The only way to reconstruct when each is meant is by reading Genesis 1, understanding the proper sequence, then adjusting the verbs in Genesis 2 to fit that sequence.
But this is not the case. The Genesis 2 account has no trouble referring to an event that has already happened. Genesis 2:8 says, “there he put the man whom he had formed.” This “had formed” verb uses a different tense to clearly step out of the narrative sequence recounted by the other verbs.* Similarly, after Adam names the animals we are told that “for the man there was not found a helper as his partner” (2:20). And in solving Adam’s aloneness, “the side that the LORD God had taken from the man he made into a woman” (2:22). All three of these verses mark an action that has already occurred.** They are marked clearly, so they show up in any translation.
By contrast, the forming of the animals does not step out of the narrative sequence (as @dscottjorgenson noted, the plot depends on this being a sequential event). It uses the same tense of verb as other events in the narrative. There is no more reason to say that God “had formed” the animals than to say that God “caused a deep sleep to have fallen upon the man, and he had slept; then he had taken one of his sides and had closed up its place with flesh.”
Placing a “had” in front of some of the imperfect-tense verbs is an arbitrary exercise, as can be seen in how the ESV and NIV both insert a “had” in 2:19 about the animals, but only the NIV inserts a “had” in 2:8 about planting the garden. The NIV translators decided that both the garden and the animals needed to already exist to fit Genesis 1. The ESV translators decided only the formation of the animals needed to be adjusted. Of course, most other translations decide not to adjust either verb, giving the final say on the sequence of events to the inspired writer rather than the translators.
* It uses a verb in the “perfect” tense, as opposed to verbs in the “imperfect” with a waw-consecutive that generally indicates sequential action. But “perfect” and “imperfect” are simplifications based on grammar in our language.
** There’s a fourth in 2:5 about “had not caused it to rain”, but since it occurs before the narrative is in full swing, it’s less clear.