The ruling by a US district judge is the latest twist in a long-running legal battle going back to 2013.
Costco, which is to appeal against the decision, argues that "Tiffany" is now a generic term for the rings.
But it has now been told it must call them "Tiffany-style" instead.
The dispute centres on the sale of solitaire-style rings, comprising a diamond mounted on a single band with six prongs.
Costco, which had sold 2,500 of them, put them on display with the label "Tiffany", although they were not in fact made by the jeweller.
An earlier court ruling in October 2016 ordered Costco to pay $5.5m in compensatory damages and $8.25m in punitive damages.
But in a further court decision on Monday, the compensation was increased to $11.1m, while the amount of punitive damages still stands.
Kate Swaine, a partner at law firm Gowling WLG, said: "This damages award may seem excessive given that only 2,500 products were sold, but if it can be argued that an infringement is blatant and where it relates to such a well-known brand, the claimant is entitled to seek punitive damages.
"Brand owners will welcome this decision as an indication of the risk that third parties run in trying to make associations with famous brands."
The cheapest engagement ring on Tiffany's US website currently costs $12,000 (Â£9,300) while customers can buy one from Costco's website for only $380 (Â£295).
US wholesaler Costco is facing a $19.4m (Â£15m) bill for damages after jewellery chain Tiffany sued it for infringing its trademark by selling "Tiffany" engagement rings.