Yes, it is the identical that Sonic hedgehog protein expressed in each tissue. The tissues develop into different structures because tool kit genes have multiple roles in different tissues and cell types. The specificity of their action is determined by the context provided by the other toolkit genes that act in combination with them.
The dramatic and diverse roles of Shh genes are a striking example of the different roles played by toolkit genes at different places and times in development. The outcomes of Shh signaling are different in each case.
The Shh signaling pathway will induce the expression of one set of genes in the developing limb, a different set in the feather bud, and another set in the floor plate. The outcome of Shh signaling depends on the context provided by the toolkit genes that are acting at the same time.