To explain this to yourself, first you have to reject a common misconception.
Do batteries store Ampere-Hours? Nope. Wrong.
Batteries don't store any electric charge. Instead, batteries are chemically-powered charge pumps. The path for amperes is in through one terminal, through the battery, then out through the other terminal. Batteries are "electricity pumps," and they don't accumulate any coulombs or AH or electrons inside.
In other words, an ideal battery doesn't block the flow of charges. Instead it's a short-circuit. When connected to a load, a battery provides a complete circuit, with no beginning or end.
Doesn't this mean that, during electric currents, the battery DOESN'T provide any flowing charges?!! Yes, exactly right. The charges are provided by the conductors. Within the battery's (very conductive) electrolyte, the dissolved ions are the electric charges (and in lead-acid batteries, the charges are the acid's mobile protons. Flowing protons!) Then, out in the copper wires, the charges are the movable electrons of the copper metal.
So, what does "ampere-hour" really mean? It's a convenient way to express the total amount of coulombs that a fully-charged battery can pump through itself before being exhausted. When the battery is dead and the AH expended, it means that the chemical fuel inside the battery has been consumed, so the pumping-process comes to a halt. The ampere-hour is actually used as a measure of chemical fuel, rather than a measure of amperes or coulombs or electric charge. Fortunately the total of "chemical fuel" being consumed inside the battery is directly related to the number of coulombs pumped through. We don't need to somehow weigh the fuel remaining. Instead we can just watch the flow of coulombs.
So, in rechargable batteries, if we force the charges backwards through each cell, then the chemistry runs backwards, and the "chemical exhaust" gets converted back into fuel again. In a flashlight battery, the zinc chloride is turned back into zinc metal, and chemical energy is stored. Or in a fuel cell, the H2O gets "unburned" and forms new H2 and O2 gas. The battery is again ready to power your devices, converting the "fuel metals" back into solid "exhaust products." The fuel cell again can burn the hydrogen into water, or your Lithium cell burns the lithium metal into lithium salts.
So, notice that a "charged" battery is not full of charge. It's full of chemical energy, full of joules or KWH worth of chemical fuel. And, a "discharged" battery contains exactly the same amount of electric charge as a "charged" battery. Confusing yes! With batteries, the word "charge" refers to a charge of energy, and not an electric charge of electrons or protons. (Similar: when we "charge" a cannon, we give it a charge of gunpowder, not a charge of electricity.)
OK, original question: why do the Amper-Hours remain the same when batteries are hooked in series? It's because each battery only has enough chemical energy to pump a certain number of coulombs through itself. WHen hooked in series, the total number of coulombs don't add up, since the coulombs coming out of one terminal just goes right back into the terminal of the next battery in the chain. That means a certain number of coulombs passes through the entire chain. It doesn't increase as it goes! If one battery passes 1000 coulombs through itself, well, all the following batteries in the chain will do the same. The voltage does stack up, and so does the energy. But series-batteries pump the same total charge through themselves that a single battery does.
Will a water-analogy help. In plumbing, a "battery" is a constant-pressure water-pump that's powered by a mechanical wind-up motor, with some energy stored in the motor's spring. The spring-pump can be rated in flowrate-hours! Each waterpump can only pump a certain number of gallons before its spring totally unwinds. Stacking up many pumps doesn't alter the total gallons that the spring-motors will pump through the chain. (Stacking up many pumps will add up the pressures, which does add up the total energy produced by the chain of pumps.)
Note well that "Ampere-Hours" actually means "Coulombs-per-second times hours," which means the same as "Coulombs times 3600." One ampere-hour is just 3600 coulombs.
Second question: can we swap voltage for AH, while keeping the size of battery constant? Yes, to some extent. But cell-voltage is determined by the "corrosion voltage" where water touches conductor at the plate surfaces. You can alter the metal, and choose battery types between about 0.5V and 4V, but that's all. The voltage comes from the reactivity of the metal, and from the "aggressiveness" of the solvent action of the electrolyte. When water dissolves metal, the dissolving process is halted by the build-up of roughly 4V between water and metal, with the metal having negative polarity and the water being positive. Drop some metal into water and it dissolves furiously ...but the metal immediately charges up to ~4v negative, and the corrosion halts. Once this voltage appears, the water no longer can drag positive ions out of the negative-charged metal. The battery's own voltage is halting the corrosion of the plates. Different metals give different voltage, as do different electrolytes (such as H2O versus molten salt, molten sulfur, etc.)
So, you can have a few volts per cell, down to a few cells per volt, but nothing further. Cell voltage is limited by the chemistry, which is limited by the voltage-steps between electron orbitals in the conductive battery-plates.
On the other hand, you can make the plate-area larger and larger (from AAAA to D-cell, or far larger,) and that increases the total amount of "fuel," and increases the AH rating of each cell. Two or three volts per cell, but use infinitely-wide battery plates for infinite AH rating. Roll the plates into a cylinder for a "DDDDDDDDD"-cell with an infinite number of Ds.