I am no expert in the matter but that seems highly unlikely. There are 2 ways how to approach that I know of:
detect/exploit/sniff known interface
Like SPI,JTAG,... or even memory interface (address,data and control buses) and do the dump with some MCU to PC adapter attached to it or dump to SD card or what ever. This is not invasive approach but needs to have the interface present. If you do not have the pinout for the chip (or the PCB ...) try to find the other stuff connected to it. Also using oscilloscope helps.
However if the access to data is restricted you're out of luck (fuses,HW/SW key ...). Also this way is not guaranteed that you will got the whole memory dump either. Mostly the interfaces are controlled by software so you would need to hack in the protocol. With HW interfaces you got direct chip access according to your privileges set by fuses...
see related Thought I found serial port - broke embedded device instead! Help?
Die shot analysis
In some cases the chip is sliced to obtain a die shot (but that of coarse destroys the chip for good). Then according to used technology you can reverse engineer the whole circuit diagram including PROM related stuff and obtain the data "directly". With EEPROM is this tricky as the information is stored as charge so you would need some really tricky sensors to scan that more like some liquid crystal based solution. but the possibility of error is extremly high as the info can be discharged by the slicing, by any contact with air or anything. Better would be reverse the circuit and detect programming interface on some spare chip and then use #1 on the real one ... FLASH memories are based on domino technology so it should be detectable from the die shot directly like PROM too.
To make some universal HW for this you would need to have some analog multiplexors switching matrix to be able to support any pinout and then prior to use configure which pin is what (power supply, data, ...). That is more or less possible (considering voltage, current, impedance, and frequency requirements) but to use it you would need the necessary info you do not have for unknown chip anyway ...
To detect the chip first you would need to try power it up which will most likely burn any chip with wrongly selected pins... making universal detection impossible. For a group of similar chips is this possible but again you would need some info about them ...