Why don't all devices use this? It adds cost and complexity. Is their any other reason for not doing something?
Seriously, I'd say that there are plenty of options and implementations for this. Having two equal batteries doesn't make much sense, so often the second is used for emergency or limp-home power. For instance, your PC has a RAM retaining battery on the motherboard for when you loose power. A laptop often gives a "Low battery" warning, at which time you're welcome to reduce power however you can.
I think that your statement that 'batteries work best if they are used until they are completely drained, and then recharged.' is a little broad. This is more the case for Nickel-based (NiCd and, to a lesser extent, NiMH) chemistries. Lithium Ion cells don't suffer this memory problem. In fact, their lifetime improves if you avoid deep discharges. See this page from BatteryUniversity.com for reference.
There are a couple of options for doing more intelligent power management in your own devices.
The simplest is an ORing diode on the power supply. If all you want is a hot-swappable power supply and you have a bit of leeway for your inputs, you can connect backup battery to the anode of a diode, and connect the cathode to your main battery. When the voltage of the main battery dips to 0.7V less than your backup (Or is removed), the other battery kicks in. Be careful of leakage current into the backup battery, it might overcharge it.
Alternatively, you can use a power mux IC like the TPS110. This lets you select your input independently (or dependently, if you prefer) of the input voltages, instead of always using the higher supply.
Finally, Linear Technology incorporates what they call "PowerPath" controllers into their battery charging ICs. I've used their LTC4011 which seamlessly transitions between battery and external power, and charges the battery while running off of the external power.
It seems you are making this overly complicated. Have the battery power the Pi always. When you have wall power available, it powers the battery charger. During that time the battery is getting charged since presumably the charger produces more current than the Pi draws. If not, get a different charger.
Nothing needs to be specifically switched. Whether the battery is being charged or not, either way the battey voltage is used to run the Pi.
Best Answer
The optimal solution would be to use rechargeable battery cells that have adequate capacity to run the fish pump for the duration required. Assuming that you've already got the batteries and chargers and are trying to make this work.
You haven't told us, but I'm also assuming from your post that the battery chargers you have are designed to charge 4 cells in series, rather than individually. I'm also assuming that this is a backup DC-powered pump and that you have an AC pump that aerates the fishbowl when the power is on.
You need a line-powered DPDT relay with contacts that are rated for the larger of the charger current or discharge current. Also note that you may want to design some form of low voltage cutoff, as your pump will fully deplete your cells, likely causing cell reversal.
Apologies for the chicken scratch schematic:
When the AC power is on, the relay is energized and the batteries are connected to the chargers as independent series (1P4S x2) strings. When the power fails, both packs are connected to the pump in parallel (2P4S), increasing your runtime. Note that there are a few aspects of this approach that will lead to shortened battery life, but hopefully its enough to get you up and running.