Y-ing a power vented tankless into a common flue with the furnace/boiler presents serious safety issues, particularly (but not exclusively) if the other unit is atmospheric-drafted (which yours appears to be.) Test with a smoke-pencil or dangling thread/spiderweb to see if exhaust from the tankless is being actively back-drafted into the furnace room out the dilution air intake of the furnace/boiler when only the tankless is running. (Try the converse as well.) You may be able to tell with your nose or hand. If yes DON'T USE IT, and vent them seperately ASAP.
Also according to Section 8 on page 45 of the manual, the N-069M vent pipe needs to be category-III (stainless steel) for it's entire length, which your installation clearly isn't. This installation faux pas will result in condensation in the flue eroding masonry in unlined or tile-lined chimneys, and the lifespan of that B-vent and galvanized flue piping won't be very long. While chimney damage is potentially expensive to repair it isn't likely to present the immediate health hazard of the combined flue issue. (Though it can be, longer term.) The date of manufacture appears to be 2004 on the front label. If it's been running 4-5-6 years into a masonry or tile lined chimney that chimney needs to be inspected for damage. (Dilution air from illegitimate vent connection to the other appliance may or may not have saved the chimney from damage.)
The gas lines appear to be necked down to 1/2" flex line, which also doesn't quite cut it for a 190KBTU+ appliance. (There may be more issues than that with the total gas line length to the gas meter/regulator too, particularly but not exclusively when that furnace or boiler is running/starting during a hot water draw.) See the table on p. 47. Get out a tape measure paper and pencil, see if the distribution line is up to the total burner capacity.
The reasons for the flame-out and re-start symptoms can be several, but probably are not related to the improper flue venting installation, but the gas line issues might be (or not.)
IMHO this unit needs to be re-commissioned by a competent installer before worrying about the operational issues. However...
Beyond the tankless, it's possible that the temperature/pressure valve (the one with the yellow tag on it below the water heater) is opening up, cutting hot water pressure to the shower as it dumps most of the hot water outflow, eventually re-closing slowly as the output temp of the tankless drops.