He spends half the film angsting about whether he's a real little boy or not. Why couldn't he just check his 'eye code'?
The only reason I can think of is that maybe Nexus-9 replicants don't have eye codes any more?
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2036: Nexus Dawn, one of the promotional short films for Blade Runner 2049, shows that Nexus-9 replicants do have an eye code.
Given that K has gone through his life knowing he's a Nexus-9 replicant and works as a Blade Runner, he almost certainly has an eye code as well.
Having an eye code does not disprove that K is the child he's searching for. Sure, the child would not be born with an eye code, but the people who hid the child have already altered the official DNA birth records. For all K knows, they may have also had the ability to add an eye code to a baby replicant. (Or for that matter, they could have altered the DNA birth records to be different than the child's to even further hide their tracks).
Let's not forget too that the "Black Out" (detailed in Blade Runner Black Out 2022) destroyed a great deal of information about the past. This means that there is enough unknown information that K can never know for sure if he's got the whole picture.
Regardless of any proof that K finds, we see throughout the film that he wants to be the child. First, there is Joi encouraging him to think this way:
I always knew you were special. Maybe this is how. A child. Of woman born. Pushed into the world. Wanted. Loved.
We see too when he visits Dr. Ana Stelline, she looks at his memory of the toy horse and says:
Yes, this memory happened to someone.
Note that she says "happened to someone", not specifying whether it happened to him or to someone else, yet K assumes that it happened to him and this seems to be the turning point where he believes he is the child.
Later when K meets the hidden colony of replicants, their leader says something along the lines of "You wanted it to be you." This is clearly a tragic flaw with K: he never attempted to disprove that he was the child once he had convinced himself that he was.
Even if K found all sorts of proof indicating that he wasn't the child (including an eye code, different DNA from the birth records, serial numbers on his bones, whatever), he likely would have still hoped that it was part of the coverup to hide him, the child.