Do double-pole breakers allow the rated current or twice that amount

I think you're going to need an electric service upgrade if you want to use this water heater.

Your 100 amp service is already somewhat loaded, as well as the box being physically full.

If you call for hot water while the dryer or range is in use (as obvious heavy loads on your panel), you may well trip the main, if you do not upgrade to 200 amp service before installing this water heater.

A different water heater might make more sense, unless you have other reasons to upgrade your electric service; one of the heat-pump water-heaters (or a heat-pump-add-on to an existing water heater tank), perhaps.

Edit, Add: Unless this is a house that stands vacant for long periods of time, there's surprisingly little "standby loss" (what you save by using an instant tankless electric resistance heater .vs. a tank-type electric resistance heater that's insulated to modern standards) - and if it does stand vacant for long periods, you can switch the water heater off when you leave. Changing to a heat pump water heater (HPWH) can, by contrast, save quite a bit and get you "free dehumidificaton" in the summertime. If you have gas available, it's more of a toss-up - gas and HPWH can be similar in cost to run, and a gas water heater will usually cost considerably less to buy. Most of the cost of running a normal (modern, well-insulated) tank-type water heater is, in fact, the cost of heating the water that you actually use. Most households will see little savings from using an instant-tankless waterheater, in practice.

It sounds like you may be mistaken as to how this is wired, or that perhaps I'm just not understanding your explanation. As others have mentioned, it's not possible to get 240 volts from a single pole in a 120/240V split phase system. Each tandem breaker provides 2 120 V circuits, this is true. However, if you measure between the terminals on a single tandem breaker, you'll get 0 volts. This is because the terminals are both powered from the same leg, and so are at the same voltage potential. If you measure from a terminal on the top tandem breaker to a terminal on the bottom one, then you'll measure 240 volts. This is because each breaker is connected to a different leg, which are each one half of a 240 volt circuit.

With all that said. For this setup to work, one appliance would have to be connected to both breaker. Something like this...

Notice that each appliance circuit has one wire connected to each of the tandem breakers. In this situation, you'd need a device like Speedy Petey shows.

Which ties the breaker handles together, to provide common trip characteristics.

Notice how the inner handles are tied together, and that the outer handles are also tied to each other. This way if either trip (or are turned off by the user), the entire circuit is shut off.

If this is wired the way you've explained, where the dryer is connected to the top tandem and the heater is connected to the bottom. Then there's some magic going on in those breakers.