Of course a miniature is a small painting. But who says "miniature" means "small"?
Here's the fabulous Online Etymology Dictionary with its entry for miniature:
1580s (n.) "a reduced image," from It. miniatura "manuscript illumination or small picture," from pp. of miniare "to illuminate a manuscript," from L. miniare "to paint red," from minium "red lead," used in ancient times to make red ink. Extended sense of "small" (adj.) is first attested 1714, because pictures in medieval manuscripts were small, influenced by L. min-, root expressing smallness (minor, minimus, minutus, etc.).
In other words, "miniature" didn't originally mean "small". It meant "painted in an illuminated-book style" and, before that, "decorated in red", as in red-letter days, rubrication, all that.
As for why "red-lead pigment" was called minium: it's apparently named after the place it was naturally found in Roman times, in north-west Spain somewhere near the Minius (Minho, Miño) River. That's from Wikipedia.
So, the modern word "miniature" only means "small" because miniated paintings (ie paintings created in an illuminated-manuscript style) were themselves small. Such book miniatures were small because they were painted in fine detail using tiny brushstroked lines of gouache or tempera, which took a long time, and because illuminated books were not usually the size of large canvases, the price of calfskin being what it was. (Even more expensive now.)
However, one could in theory paint using an illuminated-manuscript style in fine detail using tiny brushstroked lines of gouache or tempera ... on an enormous ground. It would be a gorgeous paradox -- an enormous miniature.
Really, there is no good reason other than tradition and time why a miniature has to be a certain size. Admittedly, those are good enough reasons for most people most of the time, but they are arbitrary reasons too, and could be ignored.