Yes, "Thanks" has a similar usage in English. But, using the abbreviated response "Thanks" can be ambiguous in certain situations. It could easily have two meanings in your example.
The shortened reply "Thanks" can have any of the meanings of the fully written statements - "Thank you", "No, thank you", or "Yes, thank you". "Thanks" could also mean a doubly intended "Yes, but not really" or a brisk "No" simply depending on whether it is said with a sarcastic or negative tone in your voice.
When used along with appropriate contextual signals, it usually conveys the exact meaning intended like when it is said along with an anticipatory facial expression or while giving departure clues to indicate that you are leaving. The context of a dialogue usually makes all the difference.