Periodicity of complex exponential in continuous and discrete time (Eq 1.51, Signals and Systems by Oppenheim & Wilsky)
The simple answer is that
$$e^{j\omega_0t}\neq e^{j(\omega_0+2\pi)t}\tag{1},\qquad t\notin\mathbb{Z}$$
Since $t$ is a real variable, the inequality is true for uncountably many values of $t$. Equality is only achieved for countably many integer values of $t$. From $(1)$ it follows that for real $t$, the function $e^{j\omega_0t}$ is not $2\pi$-periodic in $\omega_0$, unlike in the discrete-time case.
Two functions of a real variable can be equal on a countably infinite set of values of the independent variable, even though they're not the same function.