By definition of the identity, "measuring the identity operator" always yields 1. Always. A sheet of paper with a big '1' written on it is a perfect measurement apparatus for the identity operator, and it doesn't even need to be connected to your system! It's the best and the worst observable.
If you have an operator that measures 1 in some states and 0 in others, it's not the identity. If the state "the system does not exist" is a state you are modelling, then your actual system is a meta-system whose state of space decomposes into a part "no system exists" and a part "the system exists".
However you are not wholly wrong in that one naturally can use the identity operator on a "subsystem" to get the "existence" operator in a larger system:
Suppose we have a quantum system described by a Hilbert space $H_1$ that embeds into some larger system $H = H_0 \oplus H_1 \oplus H_2$. Then the identity operator on $H_1$ naturally extends to the projection operator onto $H_1$ as an operator on $H$. Measuring this operator now tells you "how much" of your state is in $H_1$.
However, the direct sum (instead of the tensor product) there means that we are not using the standard notion of a quantum subsystem, but rather the notion of a "sector" (as it occurs most prominently in "superselection sector") - a subspace of the space of states that is interesting for some reason but is not a true subsystem.
A common example might be a system with a variable number of particles - its Hilbert space is a direct sum of spaces with fixed particle number. If you measured the projector onto one of these spaces, a result of 1 would mean that your resultant state is now a state with definite particle number corresponding to the subspace, a result of 0 means the resultant state does not include a state of this particle number. That is, the projectors onto the spaces with definite particle number approximate the number operator.
(If $H_n$ is the space with $n$ particles and $P_n$ its projector, then the true number operator is $N = \sum_n n P_n$.)