You actually can recharge alkaline batteries, but....
You can only recharge them maybe five to ten times maximum before they stop holding a charge, and you have to charge them extremely slowly or they'll leak. While the chemistry of alkaline batteries is reversible (like most electrochemical reactions), the electrodes don't re-form in the same physical shape they were in originally. This leads to parts of the electrodes breaking off (gradually decreasing capacity) or shorting out entirely (immediately destroying the cell). This means that after a few charge cycles, the battery won't hold enough charge to be useful anymore.
Additionally, you can't charge them very fast; applying too much charging current, instead of charging the battery faster, will start electrolyzing the electrolyte. With the electrolyte being mostly water in alkaline cells, this produces oxygen and hydrogen gas, increasing the pressure in the cell until it, eventually, bursts open and begins to leak electrolyte. This isn't as violent or dangerous as what can happen with lithium batteries, but it does ruin the battery.
Secondary cells (aka rechargeable batteries) have electrodes designed to be regenerated, both in terms of the choice of chemistry and in terms of the physical construction. That doesn't mean primary cells can't be recharged, but it does mean they won't recharge as easily as secondary cells.