Probability of $2$ people being together out of a group of $4$.

We are placing the four people in two groups of two people. Notice that person $A$ must be in a group with one of the other three people, of whom only one is person $B$. Hence, the probability that persons $A$ and $B$ are placed in the same group is $1/3$.

Your mistake is that you are considering only one possibility where A and B end up in the same group. What are the others?

Another approach can be direct counting. There are total of $4!$ possible arrangements, out of which $2\cdot 1$ that begin with AB and $2\cdot 1$ that begin with BA, but also $2\cdot 1$ that end in AB and $2\cdot 1$ that end in BA, so the probability is $\dfrac{2+2+2+2}{24} = \dfrac 13$.