When InDesign makes a Tint Swatch, you get the same color, different shade, but basically the same name (but with, for example, a 50% added at the end of the same name)
Interestingly, you can rename the Tint Swatch, but it will rename the Parent Swatch too. Parent/Child swatches always share the name in InDesign. But, if you have an object selected using the Child swatch color and then click on the Swatches panel menu button > New Swatch, you get the same tint color as an independent swatch that can have any new name you want. However, it is not going to be a child color anymore.
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