I daresay most groups describe the powers and abilities in plausible ways, regardless of whether they're "old school" or not. Plausible being defined, of course, in relation to the game world.
Perhaps. Perhaps not.
It would be like refusing to allow a 3.X character to take a "5-foot step" in combat, because you feel that 5 feet is just too much ground to cover in a single step.
Perhaps. Perhaps not.
Specifically, the example was that the DM didn't think that the fighter character could actually cause the giant to move. Plausibility is specifically the issue, so one has to assume that the power was not described in a way which the DM felt was plausible.
Specifically (if one goes by the OP), the DM is saying that the plausibility of the world is more important than the rules of the game. I.e., the sequence of importance is "plausibility then rules". This is the definitive "Old School" sequence of importance.
Now, in previous discussions (more specifically focused around CAGI), many 4e defenders claimed that 4e could be run in an "Old School" way. Specifically, it was claimed that
if the use of a power didn't seem plausible in a particular circumstance, the DM could veto its use. We now see that, for a number of EN Worlders, this statement simply isn't true. In both the circumstances of the CAGI discussion and the OP, the question is whether or not a character can compel a creature to move if the DM rules that it makes no sense for it to do so.
IOW, in 3e "5-foot step" can (should) be interpreted as more than one step (and is explicitly vetoed where it makes no sense, i.e., rough terrain), but in 4e that "push" still has to be interpreted as something. It is as much the player's responsibility as it is the DM's (more, I would say) to offer an interpretation that is plausible. Clearly, the OP did not offer an interpretation that the DM found plausible.
If the "push" isn't a physical push (i.e., the character is not physically forcing the creature to move), then either the creature is in some way compelled to move or it is not. This is exactly the same argument as with CAGI.
So, perhaps it is an "apparent lack of familiarity with the rules", or perhaps it is an unwillingness to allow rules to interfere with what makes sense (to the DM) in the game world.
RC