How battery level is indicated in power bank
Obviously anything that can be done in a chip can be done with equivalent discrete logic.  You just need to replicate the functionality of the chip to detect the state of charge and convert that into the logic to drive your LED display.  

Here is a good link about measuring the state of charge of a battery.  It explains that two main methods are voltage measurement and coulomb counting.  A graph of voltage versus state of charge for some lithium batteries is shown below.



The problem with the voltage method is that some batteries have a flat discharge zone.  This is good for the circuit it is driving, but bad for being able to detect the actual charge state from the voltage.  The coulomb method gets around this problem by measuring the actual current into and out of the battery and and compares that with the battery capacity in order to estimate the charge state.  You can of course combine the two methods in the portion of the charge state where the voltage graph is not so flat.

As you can see, there is a lot that goes into estimating the charge state, so you would probably be better off just using a low cost I2C capable microcontroller to read the battery chip and drive the display.