The conservation law $\partial^\mu J_{\mu\nu}=0$ would be a mere identity if $\varphi$ were a function, but it is not a function: it satisfies
$$
\varphi\simeq\varphi+2\pi.
\tag{1}
$$
The conservation law says that $\star J$, the one-form dual to the two-form $J$, is closed:
$$
d\star J=0.
\tag{2}
$$
However, this does not imply that $\star J$ is exact. The underlying manifold is topologically a circle, and the condition (1) means
$$
\oint \star J=2\pi
\tag{3}
$$
where the integral is around the circle. The conservation law (2) does imply
$$
\star J=d\varphi
\hskip1cm
\text{locally},
\tag{4}
$$
where "localy" means on any topologically trivial segment of the circle, but not globally over the whole circle. The conservation law (2) can be regarded as a differential version of the integral equation (3), which is presumably what the author meant by "It is conserved simply because the space of $\varphi$ configurations is a circle."
Now consider the second quoted statement, which I'll repeat with some context:
The charged objects are strings. On one side of the string $\varphi$ is at $0$ and on the other side it is at $2\pi$.
To construct the charge operator, we don't integrate the $2$-form current $J$ over a $2$-dimensional surface. Instead, we integrate its Hodge dual $\star J$, which is a one-form, over a 1-dimensional surface. The following paragraphs explain this in more detail.
The concept of a $p$-form symmetry is nicely reviewed in Harlow and Ooguri. In $D$-dimensional spacetime, the current associated with a $p$-form symmetry is a $p+1$-form $J$. Its Hodge dual $\star J$ is a $D-p-1$-form, and the conservation law can be written in differential-form notation as $d\star J=0$. The form $\star J$ can be integrated over a $D-p-1$ dimensional submanifold to get an operator that implements the symmetry. Ordinary local objects (which are associated with points, or $0$ dimensional submanifolds) are invariant/neutral under the symmetry (if $p\geq 1$), but an extended object corresponding to a submanifold $C$ that "wraps" around a compact dimension (for example) can be charged under the symmetry if $C$ and $\Sigma$ are linked (in the topological sense, like two rings that are linked together).
The most familiar case is $p=0$ (ordinary symmetry). Then the conserved current is a $1$-form, so $\star J$ is a $D-1$-form (which can be integrated over a Cauchy surface to obtain the charge operator), and the charged objects are particles. A $D-1$ dimensional submanifold can be "linked" with a $0$ dimensional submanifold (the particle's location in space) in the sense that, in spacetime, we can't deform one past the other without touching.
In the question, we have $D=3$ and $p=1$. The current $J$ is a $2$-form, and its Hodge dual is a $1$-form, which we can integrate it along any "string" ($1$-dimensional submanifold) to obtain the charge operator. The charged objects — the things that can be nontrivially linked with such a string — are also strings in this case. More precisely, they are strings in space, so they sweep out a 2-d surface in spacetime.
