Cleaning dishes with breast milk in the kitchen sink
Question
Is it permitted to wash bottles that had breast milk in them in the kitchen sink?
Answer
Dear …!
If it is forbidden to eat breast milk with meat only because this has the appearance of eating meat and dairy together. However, if it falls into a meat dish, even though it changes its taste, it does not make it forbidden. Therefore, it is permissible to wash it in a meaty sink. If there is actual meat in the sink, one should be careful not to rinse it with hot water, but in a time of need, when it is washed with soap that renders this milk inedible, there are many reasons to be lenient on that account.
As for rinsing it in a dairy sink, there is no prohibition at all.
Source
Shulchan Aruch (Yoreh De'ah, chapter 81, §7, chapter 87, §4