Are there batteries with separated charging and discharging ports?
Batteries don't have separate charge/discharge ports, power management units have. Either the PMU receives more energy than needed, and uses that excess to charge the battery, or it receives not enough and compensates the deficit by discharging the battery. Charging and discharging at the same time makes little sense.

For example, here's the data sheet of AXP209 PMU which some mobile phones and battery-powered computers use. It can manage power 3 inputs, 2 batteries (one rechargeable and one non-rechargeable) and provides a programmable set of output voltages via 2 step-down converters and 5 LDOs. The block diagram inside the datasheet should be a good start if you want to design a similar system.