My version, quickly, would be that he envisaged "points" that were abstractions. Whence "logical space" as came in first around 1900 (long discussion) as implied by Boolean algebra, which he also anticipated. Also "extensionality", still a scary concept for mathematics even post-Grothendieck. Sadly MO is hardly the place: the recent book by Daniel Garber on Leibniz makes the good point that his thought is a moving target, often distorted by later authors.
Edit: Since this question has survived closure, some more. If you look at the April 2004 version of the article "Sheaf (mathematics)" on Wikipedia. it says that some aspects of sheaf theory trace back to Leibniz. I put that in; no doubt it was rightly taken out. I just think it shows how far a serious discussion might lead. The codification of four "laws of thought" from Leibniz is probably an example of distortion, if hugely influential. It broke down around 1910 (Bertrand Russell round then wrote up three laws), and the extensionality implied by A = B if (and only if but that is trivial) A and B have the same attributes had to come back into mathematics by the back door, really. Parts of this question would be fruitful as new questions.