What's meant here is that you label your degrees of freedom. E.g., if you have a spinless particle in QM, you can take the three components of the position operator as a complete set of independent observables, i.e., the Cartesian coordinates ##(x_j)=(x_1,x_2,x_3)##. Here the "label" is just the index ##j## to enumerate the components. If you have a system of ##N## spinless particles, you can take the ##3N## position coordinates. Then you have a label running from ##1## to ##N##. The observables are functions of time (in the Heisenberg picture also all your operators that represent observables in quantum mechanics are a function of time, and the Heisenberg picture is the most natural one in going heuristically from classical mechanics in Hamiltonian form over to quantum theory).
Now in a field theory you have a continuum theory. The dynamical variables are fields, i.e., quantities which are functions of position and time. It gives you the value of the quantity (e.g., an electromagnetic field in terms of the field-strengths components ##\vec{E}## and ##\vec{B}##). This means here you have two kinds of labels, a discrete one, enumerating the components ##(E_j)=(E_1,E_2,E_3)## of the field components and the position ##\vec{x}## in space where you measure these components.
In quantum field theory thus the position arguments in the field operators are just usual number, because they are in that sense a kind of continuous label for the infinitely many degrees of freedom represented by these fields.
Note that time is always a parameter in quantum theory, no matter, whether it's a quantized point-particle system (applicable in non-relativistic quantum theory and unfortunately often called the "first quantization") or a many-body system of indefinite particle number (applicable in both non-relativistic and relativistic quantum theory), i.e., a quantum-field theory.
