You can only wear one pair of shoes at a time so, no.

Yes! Your feet need a break, even just changing shoes in the evening can help rejuvenate tired feet.

3 - Maybe

To me it depends on your feet, the age of the shoes in question, how much hiking you are planning to do on your "free" day in Switzerland.

Having said that I took only one pair of shoes on one RS tour (Best of Italy) as I was very conscious of my suitcase weight. While I did not need a back up pair, I would have traded of with them if I'd had them. I am a woman but wear size 11 so mine take up a good bit of room.

2 I've done most trips with one. But I dont care about how cute or appropriate to my outfit they are. I'll pick up a cheap pair of flip flops for wear in the hotel room and as shower shoes. On the GAS, the only time you are walking thats not on city streets is any optional trails in Switzerland. I used a pair of Merrell Moabs on that trip but I wasn't doing any difficult trails, so my regular shoes would have worked too.

Depends on what you’re use to. I like to give my feet a break and switch shoes. My husband usually brings a pair of waterproof walking and maybe a pair of sneakers. On the GAS tour we hiked a couple of trails in Switzerland but nothing requiring boots. So mainly what you’re most comfortable in on cobblestones. This last May we did the Best of Germany and my waterproof Brooks Ghosts developed a hot spot where I could not wear them for a long day of walking so I was very happy I had a second pair.

I vote YES

I am in the "maybe" camp. Will it be wet? How are your feet? Do you usually just wear one pair, day after day, at home or do you change every few days? How far do you walk every day? All these questions factor into my choices when it comes to shoe taking on trips.

1 YES
my feet need a break from the exact same confinement day after day. Switching it up make my feet happier :)

Yes!
I have had expensive, well-made shoes wear out very quickly on trips. Shoes I thought had a lot more life left in them, seemed to age and get uncomfortable with the daily grind of 15-20,000 steps. Have a great trip!

I go to Europe with a 2½# carryon bag that, loaded, weighs about 12#, so I've gotten pretty good at packing light. In over 170 days, 11 trips since 2000, I've never taken any other shoes than the ones I wore on the plane. Never been a problem for me, and I do quite a lot of walking.

So, I vote NO.

Just do what you want—there is no packing police outside of the TSA. If you have the room, take an extra pair. If you don't, don't. Worst case, you can always buy a pair over there.

I always take an extra pair, I guest blisters very easily even in shoes I have broken in. So for me Yes, as I could not purchase a back up easily. My husband can miraculously buy a pair of shoes and wear them immediately, so he never does, unless he needs a dressier pair.

2 maybe. But: have you worn THOSE EXACT shoes all day, covering many miles on sometimes rough terrain, with no problems? I haven't taken that tour, but the description (as I'm sure you know) talks about multiple days of "Strenuous walking: 2–8 miles throughout the day with lots of hills, stairs, and uneven terrain." (Don't you wish they would make up their minds how many miles you're likely to cover?! Huge difference between 2 and 8!) If not, you might want to consider also taking the most comfortable pair of shoes you own.

DO I NEED MORE THAN ONE PAIR OF SHOES?

Probably not.

I did 4 weeks in Italy with one pair of North Face trail sneakers.

I do not count flip-flops as a pair of shoes, but take them for the hotel room.

I found my shoes 'rebounded' overnight.

If you know they work well for your feet, I say no. I'm a woman and bring an extra pair, but I find myself wearing the same pair almost the entire trip. I am confident I would be good bringing one pair.

What time of the year?
In summer I like to have a pair of sandals too. It gets hot in Europe in summer, even in those countries.

If you walk a lot with sandals I recommend orthotic sandals with arch support, like the ones below. They are great.
https://a.co/d/0gjOCT6m
https://a.co/d/06vrhsph

For the hotel room I wear the spa hotel slippers I occasionally take from prior hotel stays. They take no room. If you don't have any you can buy on Amazon for less than two dollars (see below)
https://a.co/d/0a4xk9hO

They sell shoes in Europe if you decide that your one pair is not enough.

Ideally, for reasons suggested 1 - YES
But because you are taking only a backpack, and shoes are available for purchase - I vote 2 - NO!

2-No. If Ecco is one of Rick Steves' recommended brands, that should be enough of an endorsement. If your feet are sore after a day of tourism, perhaps your shoes don't fit properly, or don't offer adequate support. My travel shoe is a Timberland waterproof Oxford - excellent support- can wear them all day without fatigue.