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You have to remember that Bestiary creatures always list a space they take up and a reach in their stat block. Those are the only numbers relevant to how much space you take up and how far away you can reach something.
A lot of creatures have fluff text that describes them as a certain height such as 32 feet tall, but if their stat block says they have a space of 15 ft (a Huge creature) then they only take up space equal to that 15x15x15 ft cube (3x3x3 squares on the flip mat). Then, the reach tells you how far away you can hit from any point in your space (for tall huge creatures it is typically 15 feet, for long huge creatures it is typically 10 feet)
That is what the rules say.
However, if you are a GM, unless you are playing organised play, it is entirely your purview to increase the reach of a creature based on the descriptive text.
Taking a Huge Elemental for example, the descriptive text lists it as "32 feet long". However, it has Space 15 ft. and Reach 15 ft. Thus, RAW, you take up a 3x3x3 and have reach 15 ft around you.
As a GM, you could rule that it took up more spaces, but as a player, you should consult your GM since it is not RAW.
Also note huge elementals are described as "32 feet long" rather than tall.
Your player is basically correct. While I am not aware of a specific rule in Pathfinder that states the number of squares in height a creature has the general concept has always been that you are as tall as your 2d space.
So if you are 3x3 then you count as 3 squares tall.
Back in 3.5 D&D they had a general rule about this.
Natural Reach: Natural reach is how far a creature can reach when it fights. The creature threatens the area within that distance from itself. Remember that when measuring diagonally, every second square counts as 2 squares. The exception is a creature with 10 foot reach. It threatens targets up to 2 squares away, including a 2 square distance diagonally away from its square. (This is an exception to the rule that 2 squares of diagonal distance is measured as 15 feet.)
As a general rule, consider creatures to be as tall as their space, meaning that a creature can reach up a distance equal to its space plus its reach.
I am guessing it wasn't so much of a removal as a 'wasn't on the website so didn't get included in the new rules' situation like so many other rules.
Many rules that were available in the book(s) never made it onto the 3.5 srd.
If your player still insists that a 32 foot tall giant is in reality only a 15 foot cube...
Well then, height is just an illusion. If this is so, altitude is also just an illusion, so his clever character cannot, in his wierd game reality ever actually fly above the fray. So sad.
Tell your player to cut it out or you will limit his character choices to gelatinous cubes.
well in my opinion, yes it's 32 ft tall, but it only blocks its 3x3x3 cube in combat. its how the game compensates for its reach, yes its spilling over into other squares a bit visually, but from a game mechanic perspective, other people can occupy those spaces since it's not overly invested in those extra squares, but since it is flexibly in there it can reach those squares for attacking. So when it comes to things like squeezing and finding a cage big enough for those types of creatures, you most certainly should say you need a cage bigger than 3x3x3 but from a combat/reach perspective the player is correct to say it only takes up 3x3x3 for the sake of a game token (basically the space it is unwilling to share and gets special game mechanics when you try).