Bike temporarily feels like it switches into a lower gear when rapidly braking?

That is not an intentional feature and it should not be happening.

Check to see if the chain is in fact moving onto a larger sprocket under braking. Find a area where you can ride safely in a straight line while looking down between your legs at the rear derailleur, hit the brakes and see what happens. Alternatively have a friend ride behind you and observe (probably a different friend than the one who made assertions about intentional shifting).

If the derailleur is shifting, possibly what is happening is that movement of the suspended rear triangle is causing effective shortening of the gear cable which is pulling on the derailleur and shifting into a lower gear.

Gear 1-6 is a big jump - its roughly all the way up/down the cassette.

So I'm wondering if your bike's front mech is getting involved somehow. Does this happen equally when on the big chainring as on the small ring?

When you reach for the brake levers, is it possible your hands are accidentally actuating the shifters ? I would suspect not on a pod/trigger shifter, but your bike has grip shifters/twist shifters which might be reacting to the rotational input from your hands while braking.

If they were road bike brifters I'd wonder if the braking action is also rotating the shift part of the lever, but the front gears would make it harder, not easier, and the rear gears need 2~3 full swings to jump 6 gears.

This is a good puzzling question, thank you for asking it.

This seems very much to be a case of freewheel/freehub pawls (the mechanism which allows the wheel to rotate without turning the cassette and pedals) sticking and not springing back into drive mode immediatly, after coasting.

You say when braking, but it might be when coasting or backpedaling before a corner, which happen alongside braking.

It usually happens when the lubricant in there gets old, thick and sticky. Or if it's been contaminated somehow.

"I suddenly need to pedal faster temporarily otherwise I will lose my speed" - this may be a sign of some partial pawl engagement, or it just seems like engagement because of excessive friction in there.

I'd analyse this carefully with the bike in a bike stand or upside down, or have it checked at a local bike shop and get the faulty part serviced or replaced. You can do it at home, there are instructional videos on Youtube on how to service freehubs and freewheels, but you'd need special tools, and sometimes they are not worth servicing.

I don't think the bike actually shifts gears. You say the bike "feels like" and "acts as" if it is in a lower gear, and I'm pretty sure you'd have noticed the sound and feel in the pedals if actual shifts were taking place.