Factorial and exponential dual identities

There is a close relationship between the two identities, but I don’t know if the exact formal similarity is anything other than a neat coincidence along the lines of the Sophomore’s dream (although I could of course be wrong about this). First note that the second identity can be written as

$$1 = \int_{0}^{\infty} e^{-x} \frac{x^n}{n!} \, dx$$

and therefore it is equivalent to the identity

$$\frac{1}{1 - t} = \sum_{n=0}^{\infty} t^n \int_0^{\infty} e^{-x} \frac{x^n}{n!} \, dx = \int_0^{\infty} e^{-x} e^{tx} \, dx.$$

which is an application of the first identity. (This new identity is easy to prove, since the integrand is just $e^{(t-1)x}$ so it has antiderivative $\frac{1}{t-1} e^{(t-1)x}$ and the identity follows from here.)

I know of interesting explanations of the two identities separately which are somewhat related, but not another direct connection like the one above: for the first see this math.SE question and for the second see this math.SE question.