This is quite the multipart and deep question. You seem to understand the basics but just in case, I’ll recommend this website as a good overview, albeit dated in terms of current ICs and BMS competitors.
http://liionbms.com/php/index.php
Chips
This is dependent on application. For small packs like the one you’ve drawn, there are a wide variety of chips available. For larger packs, Linear (LTC6803) and Maxim (MAX11081) are the two main competing suppliers of BMS solutions. They provide the most integrated solutions for multi-cell management. The main differences from smaller packs are the higher series voltages, daisy chain communication and noise immune signaling.
Techniques
In either case, voltage monitoring, temperature monitoring and active balancing tend to be the primary needs of a BMS. Other features such as redistribution tend to be less critical and often do not warrant the added cost and hardware.
Solutions
Unfortunately, even though nearly all BMS designs have the same goals, rarely is there an all in one solution. Solutions are based on number of cells, voltages (chemistry), tolerance for loss, communication method etc. These variables are not all standardized between one project and another. Furthermore, technologies keep changing. Linear is considering replacing CAN-BUS with shielded Ethernet. ADC accuracy is increasing. Sample rates are increasing.
The tried and true method is just simple active balancing. That is extra voltage on a cell is bled off resistively. Other than that, the rest of the system is a simple matter of reading all the cell voltages efficiently.
The Future
Eventually, we will see a more robust landscape with better standardization. Companies like Elithion or Nuvation are designing BMS solutions that are nearly drop in solutions. Chip designers are getting a better understanding of customers needs and have already sought to distinguish between the different types of li-ion needs based on different industries. This will mean that we’ll start only paying for features we need. Even battery cell manufacturers are standardizing cell types. For now though, any battery design remains quite customized.
The answer, sadly, is 'it depends'.
Adding a parallel set of cells will at least halve the impedance the charger sees. It probably would be able to handle it, but you should expect longer charge times and/or higher temperatures. And on the first couple runs, I would be ready with a IR thermometer and a fire extinguisher, but I'm kinda paranoid.
Note that you said "identical but higher capacity," which isn't really possible. Higher capacity cells in the same size typically have lower internal resistance, and that can also change the dynamics of the situation. It shouldn't be a problem, but it's something that should be noted.
Best Answer
If you increase the capacity of the cells, a well-designed charging circuit should handle them just fine, increasing the charge time as necessary. Decreasing the capacity would be more dangerous because the charge current would be too big for the new cells.
Whenever this particular circuit is well-designed or not is quite impossible to tell from the photo.
Some battery packs also estimate the amount of charge remaining in the cells. For such estimation current cell voltage is not enough, because individual cells have slightly different curves of voltage vs charge. To improve the estimate, your battery pack may use a hardcoded capacity value and/or estimate it during previous charge/discharge cycles.
Hopefully, this charge estimation doesn't affect the charge process, otherwise your cells may stop charging prematurely and you will never use their full capacity.