Undead fortitude will work at 1hp but...
"Instead" is an ambiguous referent; 'drops' is poor word choice
The "instead" is not referring to the zombie's current total of 1hp.
It is referring to the 0 hp that it would otherwise be at after the damage.
What you are erroneously reading:
If damage reduces the zombie to 0 hit points, it must make a Constitution saving throw with a DC of 5+the damage taken, unless [...]. On a success, the zombie drops to 1 hit point instead of the 1 hp it has
What the rule intends to say:
If damage reduces the zombie to 0 hit points, it must make a Constitution saving throw with a DC of 5+the damage taken, unless [...]. On a success, the zombie is now at 1 hit point instead of the 0hp it was reduced to
It is possible for the Zombie to benefit from Undead Fortitude at 1hp.
If it is already at 1hp, it drops to 0hp and then attempts the save. If it fails, it remains at 0 hp. If it succeeds at the save, it goes from 0hp to 1hp. This is not a drop. Most of the time when taking damage, it will be at more than 1hp, so you can approximate this change of from more than 1hp to 1hp by calling it a 'drop'. But in the edge case that it is already at 1hp, it will go from 1 hp to 0hp and back to 1hp with a successful save, and this is indeed not a drop in a net sense. If it gets lucky in its Con saves, it could take hit after hit while at 1hp and not drop to zero. Which is, after all, the point of the Undead Fortitude feature.
Also note that the same, flawed, wording is used in the Half-Orc's Relentless Endurance feature, which should have the same resolution.
When you are reduced to 0 hit points but not killed outright, you can drop to 1 hit point instead.
In this tweet, Eric Green asks whether a Half-Orc already at 1hp and hit by a giant spider can remain at 1hp using Relentless Endurance, but then would drop to 0hp from poison. Jeremy Crawford responds "That's correct". Neither of them bat an eye at the Half-Orc remaining at 1hp, rather than 'dropping' from 1 hp to 1hp.