If two adjacent tectonic plates are moving in the same direction, is there significant interaction at the boundary? Is it just a convergent boundary if one moves faster and "catches up" with the other? The boundary types typically shown include Transform, Convergent, Divergent, and "Other" (for when there are scattered microplates, or when other complex/compound interactions are occurring)
It’s relative motion that matters. So in your case, I assume one side is moving faster than another, so you would have a right lateral or left lateral transform fault depending on which side is moving faster. It’s “significance” would depend on the amount of relative difference in motion and, as usually defined, its potential impact on man.
Note that if they are moving at the same rate there would be no fault (and therefore no boundary.)
There has to be relative movement of the two plates towards one another for it to be “convergent.” Convergent plate boundaries are where one plate is being subducted (e.g. Japan) or pushed against (e.g. Himalayas) another. Having a combination of movements is not really uncommon, but typically of secondary importance. Good example this situation exist along the San Andreas Fault where it locally bends a bit, adding localized convergent components to what is an overall transform fault system.
There has to be some kind of interaction for seismologists and geologists to detect and trace the location of the boundaries, so it all hinges on what you mean by significant. There must be some difference in movement for one plate to break away from another in the first place. In the situation you describe, it is extremely unlikely that the movement of the two plates would coincide exactly, though it might seem to for a while when the unequal motion was temporarily halted by friction. This would build up an elastic tension between the two surfaces which would one day have to be released by slippage.