Depending on how you count there is 1 or 3 Stanley Cups.

The Stanley Cup, the trophy awarded to the annual champions of the NHL playoffs, was first awarded in 1893. It was commissioned by the Governor General of Canada Lord Frederick Stanley (hence the name). He wanted there to be an annual award/trophy for the best amateur Canadian hockey team. The trophy chosen was a silver rose bowl attached to a single-tiered circular base. Over the years the trophy grew to a multi-tiered base inscribed with the latest winners of the Cup, now awarded to the best professional team in North America. Unlike most major sports which have used different trophies over the years, the NHL uses the same Stanley Cup every year … except when they haven’t.

Presentation Cup

By the 1960s the Stanley Cup (also known as the “The Dominion Hockey Challenge Cup”, or the “Challenge Cup”) had become increasingly battered and damaged after years of being manhandled by players and staff. A clone of the Cup, complete with identical bumps & bruises, was created in secret and replaced the original Stanley Cup in 1970. This new sturdier cup is, for all intents and purposes, the Stanley Cup. It was used for at least three seasons without the players or the public being aware that the original Stanley Cup had been retired. This new Cup is called the Stanley Cup but is also known as the “Presentation Cup” as it’s the Cup presented to the championship winning team.

Upon being retired the original/real Stanley Cup was moved to the Hockey Hall of Fame in Toronto where visitors can see it on display. When the Presentation Cup isn’t on the road it too can be seen in the Hockey Hall of Fame. The public interest to see both on display however created a dilemma: what do you show when the Presentation Cup is on the road? Enter the third Stanley Cup.

the Replica Cup

Fans who travel to the Hockey Hall of Fame want to see the original Stanley Cup as well as the Presentation Cup (the new Stanley Cup). To ensure that the two are always present, even when the Presentation Cup is on the road, a third Stanley Cup (the “Replica Cup”) was created. Starting in 1993 the Replica Cup has been displayed in the Hall of Fame whenever the Presentation Cup can’t be. The Replica Cup is identical to the Presentation Cup but with a few engraving mistakes corrected.

So depending on how you count there is one Stanley Cup, or there are three.

Added info: The Stanley Cup(s) are not owned by the NHL. Despite being the crowing achievement of an NHL season, the Cup is actually governed by a trust established by Lord Stanley. At any given time there are two trustees who have “absolute power” over the Stanley Cup.