I interviewed Dr. Taylor (NLT) on Bible translation
Your Question is about how Bible translators think. This is a good and normal curiosity for many Bible readers, which we should allow on the Hermeneutics site.
I was in a Bible introduction class at Moody, and our group was assigned to research the Living Bible. The New Living Translation was Ken Taylor's second translation, which he oversaw through multiple committees. He personally translated the Living, but gave me a copy of the NLT when I met with him.
I later went back and told him about the NLT, "I like your translation, but for word studies I prefer the NASB."
Dr. Taylor replied, "I agree with your analysis. If we are going to study the words, we need to have the words. But, this [NLT] is for the people to be able to understand."
That was his defense on how he translated the NLT the way he did.
I have done Bible translation from Greek
I think you are trying to ask why it may have been translated the way it was—essentially asking about the rationale behind Bible translation. You're not asking the translators directly, you're asking people with experience to explain the thinking of Bible translators in general.
Remember, different Bible translations have different purposes. Translators keep those purposes in mind to make their translations different from other translations, thus more useful to Bible readers.
The Greek text in question:
Philipians 2:6 (greekbible.com)
ὃς ἐν μορφῇ θεοῦ ὑπάρχων οὐχ ἁρπαγμὸν ἡγήσατο τὸ εἶναι ἴσα θεῷ,
The text you are boldfacing is literally "in form of God being" from "ἐν μορφῇ θεοῦ ὑπάρχων" (eno morphae theou huparchoen). You can see in there the word for "morph" or "form".
Why, then? Speculating based on experience
I have translated the Book of Revelation. I've also taught mature adults and young children, reading from Greek and translating as I read for understanding at their level. I wrestle with these questions constantly myself.
The NIV and NLT translators probably felt that "form" might diminish the Soteriological claims of who Jesus was. They probably wouldn't want readers to get any impression of "merely form of, but not really, God". I certainly know Dr. Taylor would have felt that way as his rationale behind the NLT.
NIV & NLT differences
I lived in the Grand Rapids, Michigan area almost 30 years and rubbed shoulders with Zondervan editors, which publishes NIV. Zondervan (NIV) aims more at "adult basic understanding" while Tyndale (NLT) gives a little more focus toward children.
The NIV tends to have a word in English for a word in Greek, translating "form"/morph as "nature". But, the NIV is hardly a "word-for-word" translation as the NASB is. The NLT tried to be just a little more unbound by word-for-word translation than NIV, aiming just a little more for the understanding of the reader.
Wanting children to understand, the NLT translators likely think that differentiating between "God being in the form of God" vs "God being God" splits hairs without understanding and only causes confusion.
When I talk to seven year olds, I certainly don't try to explain how "form" is different from "substance"; Tyndale House probably doesn't want to either.
Hence, NIV would naturally want to use "nature" for adult easy reading while NLT would naturally just leave it out to avoid confusion.
It came up in the discussion that the omission in NLT and alternate of NIV term "nature" are based on presumptions of the Trinitarian publishers. Both Tyndale (NLT) and Zondervan (NIV) are Trinitarian, at least in reputation (churches 'have' doctrine, publishers 'favor' doctrine). Both their theology—and the theology of their readers especially—directed their choices as a matter of course. They wouldn't and shouldn't say so in the translation notes within their circles because of their own shared presumptions. Trinitarians will expectedly translate Trinitarian. We can't fault them for that, but we should surely observe.
Similar presumptions in translation occur with the worldviews of Mandarin and English speakers I have encountered in Taiwan over the last 12 years. So back to the OP: Why? Theological presumptions are indeed a more than 0% factor. I don't want to dive too deep in that because that should be discussed on Christianity.SE. This approaches the overlap of Hermeneutics and Christianity/SysTheo.
On a personal note, I would try to use some word understandable to both children and adult readers, not just omit. The "likeness", "form", and "nature" aspect of the incarnation of Christ is a fascinating concept behind the text. This concept existed in the minds of the NT writers. I don't understand the concept perfectly myself, so I can't make assumptions on the best way to translate text that affects our understanding on that concept.
We want translation of their words to express that beauty in their ideas. None of us know how to do that perfectly. That's why Tyndale was founded by Dr. Taylor, to offer yet one more perspective/translation to assist discussion as we collectively read a text to understand more of what can never be fully understood.
In other words, I would try to limit my own theology's influence on translation and try to let the text come through so as to be the influence on my ongoing development of theology. I couldn't do it perfectly, though. No other translator can either.