Seems a bit strange that a company selling mostly sensors would design its own microcontrollers. But with such numbers of meters/chips sold, printing the name may just have been part of a contract with the chip factory. It's a pity, but that's all it is.
If you have a multimeter, you can check which voltage the chip uses to send data to the lcd. You won't harm the PH-meter by doing that.
The drawings of Samba can be very helpful.
Check 1, I do guess Black/Red wire (battery) will be gnd and positive voltage, but you've got to be sure.
Second part, check what voltage there may be between black and yellow lines.
Keep in mind there may be no difference in voltage on some yellow lines. Should one of the last two numbers displayed for example be 1, only 2 out of 7 segments will be turned on. To see all lines/segments activated one would need an impossible PH of 18.8.
Any way, once you know which voltage the LCD/segments needs, easiest will probably be to start experimenting with the LCD to check which line triggers which segment.
This is still a bit risky since I still don't know which exact LCD-display was used. I did search on 2.5 displays in the last days though and haven't found one that doesn't use a common pin and a pin per segment yet. The number of IO-lines would also perfectly fit.
Samba, if you still have your LCD, could you test it for this project ?
Looking at the board (&datasheets of other LCDs) you won't need a resistor. If you feel safer using one, each segment draws only a few microAmperes, you probably will be able to trigger a segment using a resistor of 10k or higher.
Once we know which line triggers which segment we can almost start programming.
By the way, one thing that came to mind after seeing the meter without chip is that you may... still be able to read the meter. The PH-sensor itself gives a very small voltage, the opamp translates it to a higher voltage after which it is read by an ADC in the microcontroller and printed on lcd. What we've tried so far is looking at the output of a microcontroller, but you can try to read its input as well.
Arduino has an adc as well. If you can find the right pin, it may be possible to analogread it with your arduino.