Measuring smartphone's battery State-of-Charge during charging
To start with the smartphone question: Charge management, as well as almost all other power related functionality like dc/dc conversion for the SoC, display and power switching is done with usually a single monolithic integrated circuit, colloquially known as the PMIC (power management IC). These are seldomly if ever available in low quantities, extremely specialized and optimized for low footprint area. This is not a good option for your design question.

However, charge management ICs exist in the free market that do coulomb counting. Try looking at TI's bq series chips; these range from just simple coulomb counting pulse-output chips to fully integrated charge managers with the craziest functionality. They are the go-to design choice for almost every integrated battery solution for portable electronics - with Maxim's charge management chips taking care of most of the rest of the market.