Differential Pair

Imagine a twisted pair in free space.  Here the signals couple to each other and no reference plane is present.

Now bring that pair close to a ground to which the signals are referenced.  Any stray coupling now can couple to the plane.

Now untwist the pairs and attempt to keep the stray fields from coupling to the plane.  It won't happen.  This is the case you have with a differential pair on a PCB.  Remove any planes and you will have a given impedance.   Once you bring that pair close to a plane its differential impedance changes as the "mode" changes from differential to common mode.  

For most PCBs it is not necessary to "tightly" couple a differential pair.   You could route each half of the pair as a 50 ohm line and get a 100 ohm differential pair if the signals do not couple to each other -- but the signals would be referenced to the planes.  

But, based on physical geometries on PCBs it is often necessary to push signals close to each other to accomplish routing.  It is also necessary to have power planes for other signals.  So when all considerations are taken into account, a reference plane(s) is required and the "differential" pair must be adjusted to the proper width and spacing to accomplish a certain impedance.

So to your original question, no it is not required in theory to have a plane to generate a differential pair but it is nearly impossible in a PCB to achieve this.  I think that is why you are being told to use a reference plane or planes.

Imagine a twisted pair in free space.  Here the signals couple to each other and no reference plane is present.

Now bring that pair close to a ground to which the signals are referenced.  Any stray coupling now can couple to the plane.

Now untwist the pairs and attempt to keep the stray fields from coupling to the plane.  It won't happen.  This is the case you have with a differential pair on a PCB.  Remove any planes and you will have a given impedance.   Once you bring that pair close to a plane its differential impedance changes as the "mode" changes from differential to common mode.  

For most PCBs it is not necessary to "tightly" couple a differential pair.   You could route each half of the pair as a 50 ohm line and get a 100 ohm differential pair if the signals do not couple to each other -- but the signals would be referenced to the planes.  

But, based on physical geometries on PCBs it is often necessary to push signals close to each other to accomplish routing.  It is also necessary to have power planes for other signals.  So when all considerations are taken into account, a reference plane(s) is required and the "differential" pair must be adjusted to the proper width and spacing to accomplish a certain impedance.

So to your original question, no it is not required in theory to have a plane to generate a differential pair but it is nearly impossible in a PCB to achieve this.  I think that is why you are being told to use a reference plane or planes.