They sound different to me (native English speaker). "How did I not see that?" sounds more like the speaker is surprised that they didn't see the person was lying.

Whereas "Why did I not see that?" sounds more inquisitive and analytical, like the speaker actually wonders why they didn't figure out the person was lying to them, and the speaker is trying to figure it out.

There is definitely some nuance to this, and you can, sometimes, use "Why did I not see that?" without changing the meaning.

If you're writing this, it's better to use the right word, but if you're speaking, the person or people you're speaking to might figure out what you mean by your tone.

Personally, I would say that both have basically the same meaning.