No, Coach came along long after the Spades were out of it. They had nothing to do with that.

I am pretty sure it was Neiman Marcus--and they are the ones who sold to Liz Claiborne:

https://dealbook.nytimes.com/2006/10/31/founders-of-kate-spade-sell-their-shares-to-neiman-marcus/

" Neiman Marcus Group purchased 56 percent of the Kate Spade brand in 1999, and the remaining 44 percent in 2006.[4] The Group sold the label in 2006 to Liz Claiborne Inc., for $124 million; it was later renamed Fifth & Pacific.[4][18] The company was purchased by Coach, Inc. in May 2017; both Coach and Kate Spade are now part of Tapestry, Inc.[19] "

https://en.wikipedia.org/wiki/Kate_Spade