Can you mix stain with caulk?

No, you cannot effectively mix stain with caulk. Caulk is designed to be a sealant and filler, not a porous surface for stain absorption. While you might tint caulk slightly, it won't achieve the rich, even color or grain enhancement that stain provides on wood. For color matching, use colored caulk or paintable caulk that can be painted after curing.