Can a CFT have multiple primary operators with same scaling dimension and/or spin?

Simplest example that comes to mind: a linear sigma model (the kind you encounter in bosonic string theory). Its central charge $c$ is a positive integer and there are $c$ different primaries $\partial X^{\mu}$ ($\mu$ ranging from $0$ to $c-1$). These primaries are different fields, but they all have conformal weights of $(1, 0)$ hence the same scaling dimension and the same spin.

The answer is yes: in general, it is possible to have operators sharing the same scaling dimension and spin.

However, this is a very peculiar situation, and in practice it only happens when there is some kind of symmetry in the system. For instance:

• The linear sigma model in $d = 2$ dimensions (see the other answer by Prof. Legolasov); in this case the symmetry is Lorentz symmetry.
• The $O(N)$ models in $d = 3$ dimensions, in which the "fundamental" field is a $N$-component vector, i.e. there are $N$ scalar operators sharing the same scaling dimension. The simplest example in this family is the $O(2)$ model, sometimes also called XY-model, which describes the critical point of superfluid helium.

In most cases, it is possible to choose your basis of primary operators so that the 2-point functions are diagonal, in the sense that distinct operators have a vanishing 2-point functions. This is why most of the time in CFT we are assuming that 2-point functions only involve identical operators.

But this is in fact not always possible: there are logarithmic CFTs, a special type of non-unitary CFT (discussed in $d > 2$ in this paper), in which you can have 2-point functions that mix distinct operators with the same scaling dimension.