It can't be a contradiction because it's an imprecise colloquial statement rather than a precise mathematical statement. One way of understanding what he means is that the device that "measures" the physical observable $A$ associated with the operator $\hat{A}$ can't distinguish between the different possible measurement outcomes. In this case, the state is "projected" onto the subspace spanned by all of the eigenvectors, but this is just the identity operation.
To be a little more clear, let's imagine a three-state system and an observable $\hat{A}$ (assumed non-degenerate) represented in spectral form as
$$
\hat{A} = a_1\lvert 1\rangle\langle 1 \rvert
+a_2\lvert 2\rangle\langle 2 \rvert
+a_3\lvert 3\rangle\langle 3 \rvert\,.
$$
A precise (sharp?) measurement of this observable returns one of the three eigenvalues $a_j$ as the outcome, and the corresponding post-measurement state is the corresponding $\lvert j \rangle$. Now, imagine a case where the device is imprecise in just such a way that it can't distinguish between $a_2$ and $a_3$. Then a "measurement" of $A$ returns either $a_1$ or $a_2$-or-$a_3$. If the pre-measurement state is
$$
\lvert\psi_{\textrm{pre}}\rangle = \alpha_1\lvert 1\rangle
+\alpha_2\lvert 2\rangle
+\alpha_3\lvert 3\rangle\,,
$$
then the possible post-measurement states are
$$
\lvert\psi_{\textrm{post}}\rangle = \lvert 1 \rangle
$$
or
$$
\lvert\psi_{\textrm{post}}\rangle = \mathcal{N}(\alpha_2\lvert 2\rangle
+\alpha_3\lvert 3\rangle)\,,
$$
respectively, where $\mathcal{N}$ is a normalization factor. In the second case, we have projected the state onto the subspace spanned by the "degenerate" eigenvectors 2 and 3 (where by degenerate here we mean that the detector can't distinguish the two eigenvalues; see Note 1 below).
Finally, imagine that the detector is very imprecise and therefore can't distinguish between any of the three values. Then the result of the measurement is $a_1$-or-$a_2$-or-$a_3$, and the post-measurement state is therefore the projection on the subspace spanned by all three eigenvectors, but this is just the entire space, and so the post-measurement state is identical to the pre-measurement state. This "measurement" is the same as a measurement of the identity observable.
Notes
Really, in the "imprecise" measurement schemes described above, we're really no longer strictly speaking measuring the observable $A$. We're actually measuring some other related observable. For instance, if $a_1$ is non-zero, we could construct the operator
$$
\hat{A}_1 = a_1\lvert 1\rangle\langle 1 \rvert
+0\lvert 2\rangle\langle 2 \rvert
+0\lvert 3\rangle\langle 3 \rvert\,,
$$
where the outcomes are $a_1$ or 0, and the 0 corresponds to getting $a_2$-or-$a_3$. Still, the above is what Schwinger means, although the imprecision of language is perhaps a little annoying.
This "smearing-out" or "imprecision" of the measurement device is only one possible way that a detector can be imprecise. For instance, the device can be imprecise in such a way that it decoheres the state somewhat. This is moot in this case because what Schwinger means is what I've described above, but I wanted to make sure I wasn't lying to OP by omission.
