Sigismund III's parents -- the eventual John III of Sweden and Catherine Jegiellon -- raised him as a Catholic. Or more specifically, Catherine was a Polish princess, the youngest daughter of Sigismund I of the Polish-Lithuanian Commonwealth, and she was a staunch Catholic.

Sigismund's father John was the second son of Gustav Vasa, and did not have a good relationship with his older half-brother Eric XIV, who was a Lutheran. When Eric died without heir, John succeeded him as John III of Sweden. During his rule, he tried to give Catholics in Sweden more freedom to worship and do religious work.

So Sigismund III was raised Catholic because his mother was strongly Catholic and his father was sympathetic to Catholics.