I read somewhere that obsidian- edged Aztec weaponry was possibly the sharpest weaponry ever made. Is there any merit to this claim?

The claim is true: Aztec weapons were among the sharpest made (other peoples who made obsidian-edged weapons achieved the same sharpness, so the Aztecs aren't alone at the top of the sharpness pyramid).

Obsidian and other glasses will fracture to a very fine, and therefore very sharp, edge. Non-glassy stone blades are limited by the grain size of the stone. Metal blades can't be ground to as fine an edge, either. For applications where extreme sharpness is important, obsidian is still competitive with modern cutting tools, or even superior: e.g., J.J. Dissa, "A comparison of obsidian and surgical steel scalpel wound healing in rats", Plast Reconstr Surg. 1993 Oct;92(5):884-7.

However, "sharpest" doesn't mean "best" when it comes to weapons and tools in general. Obsidian edges, while sharp, are also brittle, and the original edge will easily break off. The breaking-off will leave two very sharp edges, but with a much less acute angle than the original, and not aligned with the cutting action of the weapon. This is why stone axes for wood-cutting are often ground to a relatively blunt edge rather than flaked to a very sharp edge. Sharpness is very useful in a felling axe, but the edge needs to be able to survive multiple blows.

Stone blades can be very sharp - even stone that is mediocre for tool-making can give very sharp edges. The edge is fragile, and the whole tool/weapon can be relatively fragile. If suitable stone is readily available, stone tools/weapons can be an effective and cheap, even if short-lived, option. Short working life can be compensated for by cheapness and the ability to make replacements in the field.

The best stone (e.g., obsidian) isn't available everywhere, and could be a valuable trade good (e.g., obsidian was traded between islands in Polynesia). The earliest underground mining in Europe appears to have been for stone (flint, also very good for toolmaking) - after surface supplies were exhausted in some areas, miners followed the seams of rock underground.

The science behind sharpness of different materials is probably best asked about on a science or engineering sub, and has been asked before. E.g., https://www.reddit.com/r/askscience/comments/1m0lkr/how_can_obsidian_be_more_smooth_and_sharp_than/

There's always room for discussion, but perhaps the section on Middle America from the Pre-Modern Non-Western Warfare section of our FAQ will answer your inquiry.