The circuit breaker is to protect the wire not the lights. As long as the wire to the new LED lights is the same size it was then you can keep that breaker. If you stepped down to a smaller gauge wire and the breaker exceeds that wires ampacity then an additional fuse would be necessary to protect the smaller wire..
To do LED's correctly the need current regulation. Good LED's will have this built in, and you'll pay more. In strip or roll form many don't have any current regulation and require and additional circuitry so as not to overdrive them, heat them up and burn them out early.
Good LED's, that will last, use constant current circuitry
Cheaper LED's, that won't last as long, use a buck converter
The cheapest LED's, that can actually catch on fire, use a resistor to try and control current. Don't buy "resistive" controlled LED's..
Be careful what you buy when it comes to LED.
Resistors are a perfectly valid form of current limiting and for cheap LED products, really common. I design boards with LED's all the time and for low power and low cost the resistor is the way to go. The problem is only efficiency and brightness regulation. You're giving up both with tradeoffs between them if you use a resistor.
In this case, something like this strip can use resistors while still not consuming much power so it's not ideal but not something to particularly concern yourself with.
And to the original question if it's labeled as being for 12V...then it's for 12V (if it's a cheap crappy product that's a different problem). Don't add any more regulation - you could even cause a failure depending on how they are managing current (for example you can't have two current sources in series).
Maine, to add, a Buck is a switching DC-DC converter and if current controlled that would be ideal. "Constant Current" is ambiguous because it includes switching regulators (efficient) and linear current regulators (not efficient).