Holonomic means that the controllable degrees of freedom in a system is equal to the total degrees of freedom.
I'm guessing you might be having difficulty differentiating between states and degrees of freedom which are both simple concepts, but sometimes confused with one another.
1st comment: The [robot] system you describe has a single degree of freedom. It just translates in a single direction like forwards and backwards. But not side to side, nor does it spin. Not a very interesting robot.
2nd comment: The double integrator has two states. You can assign them as
$$x_1 = x$$
3rd comment: The system has one input, $u$, a force
So Answer: Yes, it's holonomic. There are two states (position and velocity), but only one degree of freedom and one controllable degree of freedom (forwards and backwards).
Example of a non-holonomic system: A wheel that rolls but also 'yaws' to steer a vehicle