Is it possible to play music with a PIC16F628A
No, not without any external circuitry. Some reasons:


Both do not have a DAC, so you need at least two of these (for stereo), and of at least 16 bit to get some decent quality.
Probably much more than a few 100 bytes for the PIC or 2 KB SRAM is needed to decode MP3s.
The CPU speed is not likely to be high enough to process decoding MP3s (not sure of this statement).


There exist specific MP3 shields for Arduino like Adafruit MP3 shield: