After years of reading X-Men, I noticed something that I had originally missed: if X-23 was cloned using Wolverine's DNA, does this mean she is not technically a mutant? Or does the fact that she was born with these abilities contradict the cloning?
Yes, the Earth-616 version of Laura Kinney (aka "X-23") is classified as a mutant.
Laura Kinney was created using a corrupted sample of Wolverine's DNA. The sample was decades old and a few segments - including the Y chromosome - were unrecoverable. To create a viable sample, the DNA was augmented using that of Sarah Kinney, the project director. The resulting viable sample was the 23rd attempt, thus the X-23 designation for Laura. For the most part, she is considered a female duplicate of Wolverine, although there ARE some notable differences due to the process. For instance, she has a different claw configuration than her genetic "father".
In the Marvel universe, the term "mutant" refers to any individual of the Homo sapiens superior sub-species. Since Laura has the "mutant gene" in her genetic makeup - just like Wolverine - she is still classified as a mutant despite her origin as a clone.
Another example of this would be the Cuckoo sisters, who are classified as mutants despite being
enhanced clones of Emma Frost.