It just depends on how the water is routed to each appliance. Some have them inline, so running one affects the other. Some have them separate. Some do a bit of both
If the sink is in-line after the shower, and you turn it on hot to flush the cooled water, it drops the pressure on only one supply line, and the bather gets hit with cold water.
If the sink is in-line before the shower, and you turn the tap on warm as the hot supply is already flushed of cooled water, it drops pressure in both supplies roughly equally, the bather might notice a pressure drop but no major temperature change.
If the sink and shower are on separate branches, and you turn the sink on hot to flush the line, pressure drop on one line, but not as noticeably to the bather.
If you have a hot water recirculator, you don’t ever need to flush cooled water from the hot line.
Similar, but reversed, issues occur if you use a cold-water draw, like a toilet, hose, washing machine, et cetera.
If the shower has a thermostatic mixing valve, it will compensate for some pressure drop on one supply line to keep the temperature consistent. They’re common now, but to retrofit older showers you need to break open the wall, so it’s uncommonly done short of a major bathroom remodel, if the homeowner is even aware of the technology.
It just depends on how the water is routed to each appliance. Some have them inline, so running one affects the other. Some have them separate. Some do a bit of both
It’s more complex than that.
If the sink is in-line after the shower, and you turn it on hot to flush the cooled water, it drops the pressure on only one supply line, and the bather gets hit with cold water.
If the sink is in-line before the shower, and you turn the tap on warm as the hot supply is already flushed of cooled water, it drops pressure in both supplies roughly equally, the bather might notice a pressure drop but no major temperature change.
If the sink and shower are on separate branches, and you turn the sink on hot to flush the line, pressure drop on one line, but not as noticeably to the bather.
If you have a hot water recirculator, you don’t ever need to flush cooled water from the hot line.
Similar, but reversed, issues occur if you use a cold-water draw, like a toilet, hose, washing machine, et cetera.
If the shower has a thermostatic mixing valve, it will compensate for some pressure drop on one supply line to keep the temperature consistent. They’re common now, but to retrofit older showers you need to break open the wall, so it’s uncommonly done short of a major bathroom remodel, if the homeowner is even aware of the technology.