(We say an object $a$ is an identity of $*$ if $a*b=b$, for all $b$ in the set that's being considered.)
These operations are only examples - can you find other possible answers?
$\mathbf{X_1}$: $* = \times$ (mod 2), $\dagger = +$ (mod 2).
Here, $0$ is the additive identity, and $1$ is the multiplicative identity.
$\mathbf{X_6}$: $* = \times$ (mod 7), $\dagger = +$ (mod 7). You should check these operations are closed, i.e. $a*b$ and $a\dagger b$ are in $X_7$ for all $a$ and $b$. Again, $0$ is the additive identity, and $1$ is the multiplicative identity.
The operations don't necessarily need to have a sensible interpretation, however: here's a perfectly good $*$ operation (with identity $4$) defined in terms of a multiplication table:
| $*$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 |
| 1 | 1 | 2 | 0 | 4 | 1 | 3 | 3 |
| 2 | 1 | 0 | 5 | 2 | 2 | 3 | 0 |
| 3 | 1 | 4 | 2 | 6 | 3 | 2 | 3 |
| 4 | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| 5 | 1 | 3 | 3 | 2 | 5 | 6 | 6 |
| 6 | 1 | 3 | 0 | 3 | 6 | 6 | 2 |
Natural numbers $\mathbb{N}$: We can set $* = \times$ and $\dagger = +$. In this case, $1*b=b$ for all b, but there is no identity element for $\dagger$, as $0$ is not in the set.
$\mathbf{2}$ by $\mathbf{2}$ matrices: Consider $A=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)$ and $A=\left(\begin{smallmatrix}e&f\\g&h\end{smallmatrix}\right)$. Suppose all the matrix entries are integers. We could define operations: $A*B=\left(\begin{smallmatrix}a+e&b+f\\c+g&d+h\end{smallmatrix}\right)$ (i.e. component addition). All of these components are integers, so this operation is closed. This has the identity:$\left(\begin{smallmatrix}0&0\\0&0\end{smallmatrix}\right)$.
$A\dagger B=\left(\begin{smallmatrix}a\times e&b\times f\\c\times g &d\times h\end{smallmatrix}\right)$ (component multiplication). Again, this operation is closed. This has an identity: $\left(\begin{smallmatrix}1&1\\1&1\end{smallmatrix}\right)$
Words: Given two strings of length $n$, $l = l_1l_2l_3\dots$ and $m=m_1m_2m_3\dots$ say, each comprising of two symbols $a$ and $b$, we could choose operations that act independently on $l_1$ and $m_1$, on $l_2$ and $m_2$, $\dots$ Suppose the operation $*$ returned $a$ in the ith position if $l_i$ and $m_i$ were the same, and $b$ if they were different. Also, suppose the operation $\dagger$ returned $b$ if $l_i$ and $m_i$ were the same, and $a$ if they were different. The identity of $*$ is the string $aaaaaa\dots$ and the identity of $\dagger$ is the string $bbbbb\dots$