Plumbing – Why does flushing a downstairs toilet cause an upstairs toilet to also flush

There is a clog downstream from both of these drains. If they are both at the lowest level of the house and you have a septic system, then it's conceivable that the problem is there, but you'd also see water backing up into that bathroom when you run any other showers/toilets in the house.

Most likely, the clog is in the joint in the drain lines where the shower and toilet join on their way out of the bathroom. Get a snake to fish down the drain (probably best to do this in the shower to avoid damaging the toilet bowl) and expect to go down quite a few feet before you reach the problem.

Edit: Revisiting this question, I'd agree that there's also a likely vent blockage from the bubbling descriptions. But the fact that toilet water backs up into the shower points to a drain blockage, too, and that will be easier to fix.

Your downstairs shower has a restriction of some sort in the water supply. Start at the showerhead and work backwards to find it. You need to check pressure while flow is happening - a restriction will allow full pressure when there's no flow, but will cause a large drop in pressure when there is flow.

For the easy possibilities, check any supply valves to be sure they are fully open, and check that you don't have a limed-up or otherwise overly restrictive showerhead. Other issues can be much more involved to find/resolve - my classic warhorse of a tale is the shower/tub with terrible flow on the hot water side, due (found after I reached the point of disassembling pipes, because something was clearly very wrong with them) to one elbow being 95% full of solder, leaving only a tiny hole for the water to pass through.