You are absolutely right that ancient civilizations did mass produce certain objects. Examples are plentiful : coins, ships, swords, spearheads and armor (especially for Roman armies) , bricks, stone slabs (for Egyptian pyramids) etc ... Production was never completely "biological", coal was used widely, also molds, watermills and windmills etc ...
What makes the difference is interchangeability of parts. For object that doesn't have parts like a coin for example, nonuniform is not that important. As mentioned in other answers, ancient coins were visibly non-identical. However, aside from shaving some gold or silver from the coin, they were usually counted as such. However for objects that have at least two parts like spear (spearhead and shaft), things get interesting. Do to inability to produce them completely uniformly, you could not just take any shaft and any spearhead and join them. Usually, wooden shaft was wider than spearhead, so it would have to be filed and grinded down until perfect match was achieved. In some designs it would be secured with iron nails.
As we can see, this lack of uniformity hampered mass production of any complex design like for example mechanical clocks .To put it simply, you could not just take faulty gear from the clock and replace it with another of the same size and shape. It would have to be matched to the clock, which drove the cost of production and maintenance. Although various clock-makers and other manufacturers wanted to make things more uniform, they could not achieve this in earnest until 19th century (or even early 20th century) . Therefore, each object produced was in a sense unique.
Only after tolerances of produced parts became low enough to achieve interchangeability mass production of more complex designs became possible. And this in turn could not be done with production based on unassisted human labor. Uniformity was achieved only when machines took over dimensioning and shaping (cutting, molding ... ) of an object. Also, late 18th and 19th century saw the rise of metrology, scientific discipline concerned with measurement. This lead to standardization of measures, first in one country (France, Britain ...) , than on the world stage. Standards and etalons were established and latter more and more precise instruments to measure basic physical quantities like length, weight etc ... This in turn allowed uniform measurement of above mentioned tolerances and first forms of quality control.
Finally, it should be noted that one human activity where ancient civilizations did achieve reasonable level of uniformity without machines was construction, or actually brick-making and stone-cutting. This was especially true for civilizations like Rome, Greece, Egypt and China were we could find relatievly uniform bricks, roof tiles, stone slabs etc ... And this is the reasons why those civilizations built large cities and left plenty of constructions we could admire even today.