Unclear
A strict RAW (rules-as-written) reading of the requirements for the Defense fighting style would definitely suggest you can gain the +1 to your AC while wearing leather armor. The question is whether this bonus will also apply to your "natural armor" AC calculation.
There's nothing in the core rules to suggest otherwise. But there is a ruling on a similar issue that muddies the waters. The following exchange occurred on twitter:
Dave Williams: Lizardfolk natural AC implies you can use it while wearing armor. The +1AC bonus from magic armor says it applies to you while wearing the armor. So Lizardfolk with 13+Dex+1 AC if they are wearing +1 Leather?
Jeremy Crawford: The AC bonus that magic armor confers to you while you wear it is a property of the armor. That bonus doesn't apply to other AC calculations you might have.
This may be significant, because the wording for the AC bonus of +1, +2, or +3 magic armor is remarkably similar to that of the Defense fighting style.
You have a bonus to AC while wearing this armor. (DMG, p. 152)
This might suggest that your +1 to AC only applies when you are calculating your AC via your leather armor, not to the calculations you make when using your natural armor.
However, it's worth noting that a twitter exchange in response to the tweets above suggested that we shouldn't be too quick to extrapolate to other cases from that answer.
Dave Williams: Regarding the intent for Lizardfolk using natural armor while wearing magic armor does that same intent apply to magical armor properties other than the AC bonus?
Jeremy Crawford: My answer was only about +X armor.
Ultimately, your DM may need to apply their own judgment here. Jeremy Crawford even suggests this in response to the magic armor question.
Jeremy Crawford: If a DM wants the AC bonus conferred by magic armor to apply to the natural armor of lizardfolk, nothing in the game will break. And that DM is making a reasonable call based on the wording of the armor's description. I shared our intent, but the DM decides.