Yep. “Garbage” is an uncountable noun, like “milk” or “water.” So you might say, “Where’s the milk?” or “Where’s the water?” But you’d never say, “Where’s a milk?” or “Where’s a water?”
“Garbage can” is a countable noun, like “milk jug” or “water fountain.” So you might say, “Where’s the milk jug/garbage can/water fountain?” or you might say, “Do you have a milk jug/garbage can/water fountain?”
But saying, “Do you have a garbage?” applies an inappropriate article to an uncountable noun.
The construction seems to turn “garbage” into shorthand for “garbage can,” which is a countable noun. But I’d never heard of it before this thread.
A garbage, or the garbage or trash, sure that’s how we say it. Why the need to refer to the container? Can to me means a metal container, ours are plastic.
We also have a dryer, a washer, a fridge, a freezer, a shower, a stove. We put clothes in the laundry, not laundry hamper or laundry basket, just laundry.
We don’t say drying machine, washing machine, refridgerator unit, shower stall, you get the idea.
Note that all over those except one are verbs with -er or -or attached to the end, showing that the object in question is a performer of the verb (“fridge” is short for “refrigerator”). The stove is the exception, and it describe the object directly; the functional parts of the stove are the burners, which follows the pattern.
So your point would hold if we called it a stinker.
Calling a garbage receptacle a “garbage” is referring to it by the name of the stuff you put in there. Put the whole sentence together: “May I put my garbage in your garbage?” It makes no sense.*
You don’t put dryer in a dryer or fridge in a fridge or stove in a stove. You put clothes in a dryer, food in a fridge, and things to be cooked in a stove.
That is, it doesn’t mean what they think it means.
Actually I would say “get me a milk”, or a water or a pop or a beer. I do not bother with the redundancy of refering to a glass of milk or glass of water, a can of pop or bottle of beer. I am able to assume that the milk will not just be poured into my hands but will come in a container.
The container then becomes what it holds. A glass of milk is a milk, a can of pop is a pop, a bottle of beer is a beer, and …wait for it… a container of garbage is a garbage.
Why ask ‘where is your garbage can?’ when the recepticle is probably not a can at all? Where’s the garbage? Do you have a garbage? Sounds fine.
We’re not talking about “the garbage” or “your garbage”–we’re talking about “a garbage.” The former two sound fine to me, while the lattermost sounds weird. And to insist that “garbage can” is incorrect because the container probably isn’t metal… Well, I certainly hope you refuse to use any other remotely idiomatic speech. (For instance, you’d better not ever “turn off a light” by flipping a switch.)
Plus, if you actually look up the definition of “can,” you’ll see:
Usually metal, typically cylindrical, but not necessarily either.