It means that two things are true.

First:

$$P(X_1 < t) = P(X_2 < t)$$

for all real numbers $t$$X_1$$X_2$

Second:

$$P(X_1 < t) = \frac{1}{\sigma \sqrt{2 \pi}}\int_{-\infty}^t e^{\frac{(x - \mu)^2}{2 \sigma^2}} \,\text{d}x$$

for some fixed numbers $\mu$$\sigma$$X_1$

This doesn't imply that $(X_1, X_2)$

(*) Given the first condition, this implies that the distribution of $X_2$

I think "common" here just means that the marginal distribution $\text{N}(0,1)$ is common to both random variables (i.e., they have the same marginal distribution). Although technically this is insufficient to give a bivariate normal distribution, I think the writer probably intended that form:

$$\begin{bmatrix} X \\ Y \end{bmatrix} \sim \text{N} \Big( \begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 & \rho \\ \rho & 1 \end{bmatrix} \Big).$$

That specification would yield common marginal distributions $X \sim \text{N}(0,1)$ and $Y \sim \text{N}(0,1)$. If I were you, I would suggest noting this technicality, and then proceed on the basis that the random variables are bivariate normal. You might want to note the issue again as a caveat once you give your answer.

The exercise is badly phrased. I suspect what is meant is that the two random variables are jointly normal and have a common distribution. If they're separately normal but not jointly normal, then you don't have enough information to answer the question. If my suspicion is right, then the exercise should have said they are jointly normal.

To have a "common" distribution simply means they both have the same distribution. Thus:

$$\begin{align} \require{cancel} & \left[ \begin{array}{l} X_1 \\ X_2 \end{array} \right] \sim \xcancel{\operatorname N\left( \left[ \begin{array}{l} \mu_1 \\ \mu_2 \end{array} \right], \left[ \begin{array}{cc} \sigma_1^2 & \rho \sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \sigma_2^2 \end{array} \right] \right)} & & \longleftarrow \text{ not common} \\[10pt] & \left[ \begin{array}{l} X_1 \\ X_2 \end{array} \right] \sim \operatorname N\left( \left[ \begin{array}{l} \mu \\ \mu \end{array} \right], \left[ \begin{array}{cc} \sigma^2 & \rho \sigma^2 \\ \rho\sigma^2 & \sigma^2 \end{array} \right] \right) & & \longleftarrow\text{ common} \end{align}$$ $X_i\sim\operatorname N(\mu,\sigma^2)$$i=1,2,$