Is it possible to using arduino and uc3906n IC make a battery management system? monitor and control the lead-acid battery.
That would depend on what you want the battery management system to do.
"Monitor and control" are very general terms.
Are you completely familiar with the use of the uc3906n and with the process of managing lead-acid batteries.
It is not reasonable to expect advice about those subjects here.
If you are familar then you need to tell us exactly what role you envisage for the Arduino.
...R