How can I ask my manager for more responsibilities?

How can I ask my manager for more responsibilities without making it seem like I’m unhappy with my current role?

There is nothing wrong with asking: it's called "career planning" and perfectly normal; at least it should be. It's unrealistic to assume that you will stay in your current role for the rest of your work life, so a constructive discussion about where you want to expand and what the options and steps could be is perfectly healthy.

Many good managers will bring this up pro-actively once a quarter or so. If your one doesn't, you should do it yourself. It helps doing some homework up front: Analyze what you are currently doing, what you like about it and what you don't, what you are good and what you could better and create some ideas or guardrails about the "future state". Make sure you are fully up to speed on your past performance reviews and feel free to bring them up if needed.

The approach your manager and ask about career planning. You may to take a hint on how the react. Most better managers will be perfectly comfortable with the ask and engage constructively. If the manager reacts negatively or evasive and then you have your answer: this is unlikely with them and so you will need to look elsewhere for advancement (could be inside or outside).