Is it common that a byte in an EEPROM can be changed during using it in an equipment?
You mean a change on its own without an external write? No, this is not common.
You can sometimes see some degradation in very old equipment (like from the 80s). But with more modern devices it basically never happens.
Question rather seems to be about normal use. EEPROM is generally used to store settings or something like that. So it's contents will change during normal device operation. Although there are some uses when data does not change, like EDID EEPROM in monitors or storing serial number and config in say USB/RS232 adapters.
Is it common that a byte in an EEPROM can be changed during using it in an equipment?
Can it be changed? Sure. Just write into it.
Whether that's expected or normal for the equipment is impossible to say without knowing how that equipment in question functions. But given that EEPROMs are usually used to store configuration or user settings, they tend to be written to as well, kinda by definition.
Their very function in many cases is 'non-volatile ram' and as such they can be changed often. They have data retention for something like 100 years for the newer stuff and maybe 20 years for the older dies. HOWEVER, there is a finite number of write cycles to any specified location. This can be as few as 10,000 and often specified at 100,000 so one must be careful about how often a particular location is written to. Case in point, we had some old video terminals that stored parameters in a very early version of eeprom, one with a couple hundred write cycles max. We found that with age the eeprom was no longer 'non-volatile' and with a power outage the settings would become corrupted. When the terminals were new they could survive power cycling without corruption.