They are roughly the same size but are slightly different shapes. The barrel on the M&P BG38 is a little thicker so if your holster is form fit to a J-frame, it won't work with a BG. The BG frame itself is a little taller than the J-frame.The rear of the frame where the laser and cylinder release is located is completely different in shape so if your holster has a strap or something it definitely won't work.
To answer your first question - NO, they are not exactly the same size. To answer your second question - Maybe, depending on the holster.
My BG38 did not fit my Kydex Blackhawk J-frame holster and did not fit my leather OWB J-frame holster, but it did fit my Galco J-frame pocket holster and my cheapo IWB leather J-frame holster.