-_- convergent losses of genes are far more common than convergent gene generation because many different mutations can result in a given gene becoming inactive. The fact that the mutations on the genes which render them inactive are not identical, and perhaps also that the inactivation is spotty among organisms in those separate lineages is how we can tell the inactivation wasn't shared by the common ancestor of these organisms. Happens to humans that have non-inherited hemophilia (yes, some people with hemophilia have no family history of it whatsoever, and severe forms of hemophilia can have clotting factors not produced).
Here's a relevant paper on this gene in particular
Evolution of the Alx homeobox gene family: parallel retention and independent loss of the vertebrate Alx3 gene
It's neat that all 3 are missing the gene entirely, but despite similar mutations that resulted in the loss, the lineages are very spotty. This source mentions that zebra finches still have the ALX3 gene, while chickens do not, meaning that the loss had to occur after birds already existed. So, yeah, you haven't found anything that would confuse genomic comparisons being used to establish which groups are more closely related to each other. That is, it is pretty common for distant lineages to lack the same genes. I mean, by default, organisms do not have the majority of genes that have ever existed.
Still waiting on that 100 identical base pairs in a row. You aren't going to distract me from it. Either drop your claim that identical genes can arise in separate lineages so frequently as to interfere with using genes to establish relatedness, or provide actual evidence for it.
For your claim to be true, we'd have to observe organisms that we DON'T consider to be particularly closely related having identical genes which they could not have feasibly retained from their most recent ancestor. Your claim was not based on gene loss, only gene presence, so your attempt to derail our conversation will be as unsuccessful as it is irrelevant. Convergent gene loss is not the same thing as convergent gene evolution, and neither of them are the same as two independent lineages developing identical genes.