'for Verb-ing' followed by a preposition

Firstly, you have to realize that the purpose of a dictionary is to describe the word in such a way that people can understand the object associated with the word. A "sink" must contain the objects being washed or it wouldn't be a sink. The inclusion of "in" is helpful in conveying this and removing it would impact the effectiveness of the definition:

1) a large open container [...] that you use for washing dishes in

2) a large open container [...] that you use for washing dishes

The "large open" modifier implies that you put the dishes into the sink but including "in" makes it much more explicit. Otherwise, you could have a large open container that washes dishes akin to a washboard. While it wouldn't make a whole lot of sense, the dictionary is specifically trying to associate the preposition "in" with "sink" because that is how you use a sink: You put dishes in the sink in order to wash them.

For common, everyday usage you could remove "in" and no one would be confused but this pattern is common in dictionary entries.

"I bought a pair of chopsticks to eat rice" could be misinterpreted to mean that your chopsticks are the things eating rice. "I bought a house to live" could be misinterpreted to mean that without a house, you would die. Similarly, the sink is not washing your dishes.