The volumes of copper wire are not required to be equal in the primary and secondary, for theoretical or practical transformer operation.
However, copper is lossy, and any space in the 'iron window', the space in the core available for winding copper that isn't copper (so air, insulation, core former, tape) is making the copper thinner than it could be, so losses higher, and the transformer less efficient.
In a well designed transformer, the copper area in primary and secondary will both tend to be the same at around 25% to 33% of the iron window, the 33% to 50% balance being wire insulation and wasted space. However, if one winding needs very thin wire (so relatively thicker insulation), or very high insulation levels, or multiple secondaries, or space for a winding shuttle to fit through, then less copper area would be used.
The thing determining current in a transformer is heating, the allowable operating temperature for the insulation. While 12mm^2 wire is just about OK for 100A in a single strand in free air, this winding will be packed into a small space, together with heating from the primary and core losses. You will be able to draw 100A from it for a while, maybe a minute, certainly seconds, but not for 10s of minutes, before it gets too hot.
I recently built what sounds like a similar transformer, and used 32mm^2 (8 by 4mm^2 in parallel) for my 100A continuously rated secondary. It was threaded through a 600VA toroidal core after secondary removal, so 80% of the winding length was in fresh air.
I'm going to go out on a limb and guess your 1000VA transformer is from a microwave oven? If so, be aware that the core and primary losses are terrible, and forced air cooling is mandatory. If it is indeed a 'proper' core, better still a toroid, then your secondary will benefit from the lower heating and better cooling, and you'll be able to use 100A for longer.
Don't worry about the primary volume. If a manufacturer put it there as a transformer primary, it's going to be far better than anything you could modify.
For the highest efficiency use the thickest copper with the thinnest insulation, and fill the remaining space in the window. In practice, this means magnet wire, not plastic insulated 'wiring' wire. It will run at a higher temperature as well. Plastic works, it's a lot easier to wind by hand without damaging the insulation, but is less efficient.
Best Answer
A transformer has several failure modes, all to do with the insulation.
1) It can punch through promptly, due to overvoltage. At these low operating voltages, this is only likely to happen if there's a very large mains spike, and there's not a lot you can do about it. Transformers designed for mains connection have to be designed to survive 1500 V spikes as a minimum.
2) It can degrade over time. Keeping the operating temperature down to something you can put your hand on will generally mean this process is slow enough not to be an issue in your lifetime, and is rarely an issue in low voltage transformers like this.
3) It can be damaged promptly with heat, aka smoking or bursting into flames. This is the usual way we damage transformers, when trying to push them beyond their design ratings.
It's not the current, or the current density that kills the transformer, but the temperature. As the temperature can take seconds or minutes to rise, this means you can safely draw excessive current from transformers for short periods of time, as long as the maximum insulation temperature is not exceeded. It also means that if you're going to do a soak test to estimate the maximum safe current, you need to run it for an hour or two, to let it reach its final temperature.
While it's fairly easy to measure temperature rise, it's not possible to know what grade of insulation is used in the transformer, premium transformers may use higher temperature capable insulation. Fortunately heat goes as current squared, so you don't lose too much possible power by making a guess. I'd be comfortable with a 50°C rise above a 25°C ambient, but if you took a chance and went for 75°C rise, you'd only be looking at sqrt(3/2) or roughly 20% more power.
Measure the resistance of a winding before you start. The tempco of copper is around 0.4% per C, or roughly 10% increase in resistance for 25°C rise in temperature. Run at power for a few minutes, disconnect and measure the resistance. Repeat the cycle until you decide the temperature has stopped changing, or you stop the test at that power because it's going to get too hot. This will allow you to determine the power rating of the transformer.
It has another rating, the regulation. On load, the output voltage of a transformer will sag by a few percent. You have to be happy that this sag is not excessive for your application, a typical figure is around 5% or so. This is rather quicker to measure. Even if you're happy with a large sag, you still have to operate the transformer within its power rating.
I've not discussed voltage overload. Don't. Unlike current overload, there is no margin for going a bit higher on the rated voltage. Don't.