Is prime order group used in ECDSA unique? In ECDSA, we use a prime order group $\langle G\rangle$ for cryptographical use. Assume $\#\langle G\rangle = p$. Is there another subgroup in the curve used for ECDSA whose order is also $p$?
This question has two faces;

The number of rational points of the used curve is prime like secp256r1, i.e. $\#C(K) = p$ where $K$ is the field where curve $C$ is defined;
In the group theory, let $G$ be a finite group and $H$ be a subgroup of $G$.
Then the order of $H$ must divide the order of $G$. This the Lagrange's Theorem in group theory. More formally;

Lagrange's theorem - If $H$ is a subgroup of a group $G$, then ${\displaystyle \left|G\right|=\left[G:H\right]\cdot \left|H\right|.}$

Below a corollary to the Lagrange theorem;

If the order of the group is prime then there are no subgroups other than the trivial subgroups.

Therefore, an order $p$ group has no subgroups other than the trivial subgroups which are the identity group and the group itself.

The number of the rational points of the used curve is not a prime,  $\#C(K) = n$;
This really depends on the group. The Lagrange theorem doesn't guarantee the existence of the subgroups if an integer divides the order of the group.
Consider the Curve25519 which has a prime order subgroup $n$ and the co-factor is $h=8$ ( $h = \#E(\mathbb{F}_p)/n$) then we have the subgroup of order $2,4,8,n,2n,4n,8n$, i.e. we have all possible subgroups.
You can find a group that has two prime subgroups of order $p$ the if there are no other subgroups the curve group will be isomorphic to $\mathbb{Z}_p \oplus \mathbb{Z}_p$. Is this possible in Elliptic Curves?
I haven't seen an example in secure elliptic curves. The reason is simple; we want a large prime subgroup order that is close to the curve order in order to effectively use the curve. If there are two different subgroups of order $p$ then the curve order will be $\geq p^2$. This means you are wasting your computations.



More into theory (4.1 is in the second edition of Washington's Book);

Theorem 4.1: Let $E$ ben elliptic curve group over the finite field $\mathbb{F_q}$. Then $$E(\mathbb{F_q}) \simeq \mathbb{Z_n}
&gt; \text{ or } \mathbb{Z_{n_1}} \oplus \mathbb{Z_{n_2}}$$ for some
integer $n \geq 1$, or for some integers $n_1,n_2 \geq 1$ with
$n_1|n_2$.

If the $E(\mathbb{F_q})$ falls into the first case implies that the subgroup order $r$ must divide $r|n$. But the subgroups of $\mathbb{Z})_n$ are unique. Therefore there cannot be two subgroups that have the same order.
The second case is more interesting [1] [2] [3] since it is a direct product of groups. Since $n_1\mid n_2$ there is a subgroup of $H$ of  $\mathbb{Z_{n_2}}$ such that it is isomorphic to $\mathbb{Z_{n_1}}$. Now we have two subgroups of the same order $\Bbb Z_{n_1} \oplus \{0 \}$ and $\{0\} \oplus H$.
As said before we don't prefer curves that can have two different subgroups of order large prime.