I Is There a Connection Between Conjugation and Change of Basis?

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Can the adjoint representation of a Lie group be regarded as a change of basis?
For transformations, A and B are similar if A = S-1BS where S is the change of basis matrix.

For Lie groups, the adjoint representation Adg(b) = gbg-1, describes a group action on itself.

The expressions have similar form except for the order of the inverses. Is there there any connection between the two or are they entirely different concepts?
 
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A and B are also similar if there exists an invertible S such that A = SBS^{-1}.
 

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