That maybe what Taiwan see's as its waters (176km is the current distance of the ship from shore), but internationally recognized distance is 12 miles per the United Nations Convention on the Law of the Sea.
Those are territorial waters (12 nautical miles from shore).
The exclusive economic zone (EEZ) is 200 nautical miles from shore. Foreign ships (not warships) may pass through the EEZ but may not use the waters for such things as fishing and drilling.
It sounded like the Chinese ship was loitering in the EEZ instead of simply passing through it. Since they are enemies, perhaps the surveilance ship is considered a war ship?