We found what may have been the archaeological remains of an aquarium in the remains of Piper's Old Corner Bar in Virginia City, Nevada. The artifacts were apparently part of the debris from the 1875 fire.

Kelly Dixon (now at U of Montana), who directed the excavation, wrote about it in her book Boomtown Saloons (2005). Here is an excerpt (65-66):

Marine aquariums developed during the 1850s, and the ensuing social craze over them led to their placement in almost any home or enterprise that attempted to boast of affluence and modernity. For the first time in history people viewed the undersea world in cross-section instead of merely looking down into the water and imagining life under the sea. An 1873 article from a Carson City, Nevada, newspaper touted one of these early aquariums in San Francisco: "It is difficult to describe the novelty of the sight. There are three marine tanks already filled - one with crabs, etc.; second with perch, flounders and a variety of small salt-water fish; and a third with star-fish, etc.... Our reporter never before realized the beauty of a fish, and the wondrous grace of its motions... one also has a good opportunity to watch and study the habits of the fish while close at the bottom of the sea."

The artifacts from Piper's Old Corner bar consisted of shells, bits of coral, and part of a crab claw. There were also bits of flat glass in the same location of the site. The marine artifats appeared as though they may have been exhibited underwater, but that could not be clearly determined, and no primary source evidence confirms that Piper had this sort of attraction in his establishment.

Kelly cited an article by Stephen J. Gould, "Seeing Eye to Eye," Natural History (July-August 1997) 16-19, 60-62.

The Romans developed fish ponds, both for food and amusement. Pliny in Natural History mentions several stories of strong attachment between people and their 'pet' fishes. This passage probably refers to eels:

At Bauli, in the territory of Baiæ, the orator Hortensius had some fish-preserves, in which there was a murena to which he became so much attached, as to be supposed to have wept on hearing of its death. It was at the same villa that Antonia, the wife of Drusus, placed earrings upon a murena which she had become fond of; the report of which singular circumstance attracted many visitors to the place. Source

And this one is about dolphins, an animal which Pliny was quite interested in:

In the reign of the late Emperor Augustus, a dolphin which had been carried to the Lucrine Lake conceived a most wonderful affection for the child of a certain poor man, who was in the habit of going that way from Baiæ to Puteoll to school, and who used to stop there in the middle of the day, call him by his name of Simo, and would often entice him to the banks of the lake with pieces of bread which he carried for the purpose... At whatever hour of the day he might happen to be called by the boy, and although hidden and out of sight at the bottom of the water, he would instantly fly to the surface, and after feeding from his hand, would present his back for him to mount, taking care to conceal the spiny projection of his fins in their sheath, as it were; and so, sportively taking him up on his back, he would carry him over a wide expanse of sea to the school at Puteoli, and in a similar manner bring him back again. This happened for several years, until at last the boy happened to fall ill of some malady, and died. The dolphin, however, still came to the spot as usual, with a sorrowful air and manifesting every sign of deep affliction, until at last, a thing of which no one felt the slightest doubt, he died purely of sorrow and regret. Source

The sources seem to indicate this was more of a habit of the rich, which not all people agreed with. Cicero in one of his letters, for instance, makes fun of people who get happy when they train a red mullet fish to eat from their hands.

As far as I can tell, the habit of Roman keeping fish as pets only occurred in natural or artificial ponds. I haven't seen any evidence of glass aquariums.