Is this pairing-based signature scheme secure?
The proposed digital signature scheme is not secure! More precisely, it is not existentially unforgeable under an adaptive chosen-message attack.
Let's consider the following efficient adversary $\mathcal{A}$: it queries the $\mathsf{Sign}_{sk}(\cdot)$ oracle for the digital signatures on $m_1,m_2$, where $m_2:=m_1+1$. The received signatures are $\sigma_1=g_1^{x_1m_1+x_2}$ and $\sigma_2=g_1^{x_1m_2+x_2}$. Now, $\mathcal{A}$ computes $\frac{\sigma_2}{\sigma_1}=\frac{g_1^{x_1m_2+x_2}}{g_1^{x_1m_1+x_2}}=g_1^{x_1(m_2-m_1)}=g_1^{x_1}$. Once $g_1^{x_1}$ is recovered, $\mathcal{A}$ can also compute $g_1^{x_2}=\frac{\sigma_1}{(g_1^{x_1})^{m_1}}$.
The knowledge of these $2$ points ($g_1^{x_1}$ and $g_1^{x_2}$) enables now $\mathcal{A}$ to forge signatures on any message of their choice. Namely, given any message $m$, $g_1^{x_1}$ and $g_1^{x_2}$, the valid signature on $m$ can be computed as $\sigma=(g_1^{x_1})^{m}g_1^{x_2}=g_1^{x_1m+x_2}$. The adversary $\mathcal{A}$ outputs $(m,\sigma)$ as a valid forgery and wins the security game.
This means, that the adversary can win the existential unforgeability security game with probability $1$ already after $2$ signature queries.