Off the top of my head, I know that Pre-BCE(maybe 1000 or 2000) rice paddies in Asia kept fish in the water. The fish provided not only a ready supply of protein, but also proved to be beneficial to the harvest, as they would eat insects that harmed the plants [1]. As time went on, the fish began to mutate and take on colors. Through selective breeding, many brilliantly colored koi started surfacing in markets and were prized by the wealthy, who proceeded to keep them in ponds as decoration.
So at this point, fish are kept for their aesthetic qualities and we could basically say they're being kept as pets.
Unsourced but relevant: Song Dynasty(started around 960 CE) Chinese kept fish in "tanks" / urns as pets. So if koi ponds don't count, that might.
[1] : http://www.scidev.net/global/farming/news/fish-and-rice-flourish-together-in-paddies.html
I do not know it they were first, but the romans did have fishponds, and at least by the late roman republic the practice to have fish as pets was certainly already established.
A famous example is that of the orator and senator Hortensius (a contemporary and rival of Cicero), who did feed his fish by hand and was said to "have wept for the death of a favorite murena".
Fish keeping for adornment is an old practice. The Chinese and Romans kept fish in pools for decorative purposes. These practices, however, are probably more akin to putting a gold fish in a bowl or a pond.
The more modern practice of developing an aquarium (an ecosystem involving fish and plants) began in the early-mid 19th Century in Europe. An early author on aquarium development, A. M. Edwards, notes the distinction:
"The great difference between a fish-globe and an Aquarium is, that in the one, we keep fish only, whilst, in the other we cultivate many beautiful and wonderful plants and animals which would not flourish if we changed the water often."
The home aquarium became popular in Victorian England (both freshwater and saltwater varieties). This was an extension of leisure and the desire to create collections of exotic artifacts. Early authors called these aquariums "Oceans In a Box;" a means of viewing the ocean without taking the train from London to the sea.
Somewhere in my attic I have a book with an essay that discusses the process of acquiring fish and maintaining aquariums. I cannot remember the book, but the Rehbock chapter below is particularly good. I have only read the first thirty pages of the Brunner book, but it seems like a decent introduction to the history of aquariums.
Sources:
Philip F. Rehbock, The Victorian Aquarium in Ecological Perspective, in Oceanography: The Past (1980)
Bernd Brunner, The Ocean at Home (2003)
What about glass fish tanks? Does anybody know how they originated?
In India there is the Vedic tale of Matsya (the fish avatar of Lord Vishnu) and in some tellings Matsya started as an ordinary fish that grew to enormous dimensions, including stages where they kept Matsya in a pool specifically for him. Not quite a "pet" but it shows to me they had some experience of keeping fish.
There also appears to be entries in vaastu (basically Hindu feng shui) that talk about keeping aquariums, but I couldn't find any actual source for this, so it might be a modern adaption.