When I was reading the derivation of the matrix condition number for linear equations on Wikipedia, $||\textbf{b}||$ can be directly replaced by $||\textbf{Ax}||$ as the non-singular square matrix is considered in the picture (which is the derivation on Wikipedia page).
Derivation of the condition number in Wikipedia
But, if we consider a rectangular matrix $\textbf{A}$, I don't think we can just substitute $\textbf{b}$ with $\textbf{A}\textbf{x}$. Then how we derive to achieve $cond(\textbf{A})=||\textbf{A}||\cdot||\textbf{A}^{\dagger}||$, where ${\dagger}$ represents the pseudo inverse here?
If two vectors are the same, so are their norms. Hence, if $b = A x$, it is always true that $\|b\| = \|A x\|$, regardless of the nature of the operator $\mathcal A x := A x$.
The condition number is much more delicate, in the sense that when a system has multiple solutions, it is not immediate how to measure sensitivity of the solution to perturbations in $b$. I recommend reading the paper:
Stephen Demko, Condition numbers of rectangular systems and bounds for generalized inverses, Linear Algebra and its Applications, Volume 78, June 1986, Pages 199-206.