No, it isn't because of privacy. It is because they get a cut of the total volume processed on the cards and, as a bonus, they get details about where you spend your money.
They aren't operating a charity. You also left our hotels, airlines, etc. United Airlines makes $$ by signing up members for the United Cards by Chase.
Starbucks has one of the most seamless payment and ordering systems I use on a day-to-day basis. I'm not sure why it's surprising they would have their own credit card to make that even easier. Heck, every big box store seems to have their own card too. They cut down on the transaction fees if it's their own card, plus extra data for them.
At one point Starbucks' digital payment system was ahead of Apple Pay in dollar volume and doing double the volume of all of google pay. I didn't know Starbucks had a credit card but it doesn't surprise me.
Interest income, plus a cut of every transaction (even if not at Starbucks). Plus added spend at Starbucks - having a Starbucks card may entice more spend at Starbucks.
They aren't operating a charity. You also left our hotels, airlines, etc. United Airlines makes $$ by signing up members for the United Cards by Chase.