Electronic – Digital multimeters battery placement

The PIC has an industry standard UART, and anything the PC can do the PIC can do as regards communicating.

The only difference between the two is the PC will most likely have an RS-232 line driver in the circuit to provide the proper +/-12V or so. The PIC doesn't have this, but adding something like a MAX232 chip will convert the output to standard RS-232 levels.

All you need to know is the correct protocols to use to talk to the remote device, and to implement them in the PIC's firmware. Note that the PIC doesn't have any hardware handshaking as standard, and a very tiny (2 character) receive buffer, so I tend to write my own send/receive routines which implement hardware handshaking using a couple of other I/O lines which also go through the MAX232 chip (CTS/RTS).

Getting the right baud rate setting on the PIC can be a little tricky, depending on which specific language you are programming in - some calculations are often required to get the divisor from the system clock speed. The data sheet is essential for this - read it thoroughly.

I expect your multimeter is rather more accurate than the grid voltage. Grid voltage can vary widely from the supposed nominal value. At least 10% should be assumed off the top, but more should not be surprising. 220V +-10% is 198-242V, so you're within spec. There are places in the world I would trust the power less than others, and frankly eastern Europe is in the second tier. It's not as bad as some places where outages are common, but more than 10% variation shouldn't surprise anyone.

Then there is the whole EU standardization attempt. Various countries had somewhat different power voltages, like 220V, 240V, etc. The EU wanted to standardize this. But instead of actually changing the voltages they just re-defined nominal with enough slop so everyone was within spec!