SIM card means different phone number?

"Imagine my surprise when I found out the SIM card changed my phone number." - Specific phone numbers are linked to sim cards provided by network providers; a sim card from a European network provider will always provide you with a new phone number, which you will receive when you order the sim.

If you want to use your US number in Europe, keep your existing sim card in your phone.

Yes, changing the SIM card basically changes the phone number associated with it. Many modern phones have eSIM functionality. Dual SIM Dual Standby (DSDS), which allows a single device to function as two different phones. You'll have to find out if your phone supports these features. I have a single physical sim phone, but I have service with T-Mobile that includes international data and text in my plan. I used to buy local SIM's but encountered the same problems as you. Since switching to T-Mobile, I can use my US number overseas without issues.

Thanks so much for these replies.
I honestly don't understand why this isn't mentioned in the RS article about using a sim card. Isn't that one of the most important things someone planning a trip would want to know? I mean, for me it was just unbelievable that reading up to be a well-informed traveler didn't prepare me for the fact that my friends, family, hotels, and ferry company would be unable to call me.
Moreover, when I got off the plane returning home, I couldn't call my ride. I had to email a relative in the US a day ahead and ask her to call him on my behalf and explain why I wasn't going to be able to call. My driver had to look up the flight info to see whether my flight was landing on time. I guess I could have reinserted the US sim card while on the plane, but I didn't think of that at the time.
On this year's trip I probably will keep my existing phone. It's what I did the last time I went to Europe, in 2016, and it worked fine without costing too much.
The info about a dual sim is interesting. I doubt that my cell phone provider company (Credo Mobile) enables this, but I can ask them.

"On this year's trip I probably will keep my existing phone." - See if it's worth changing to a similar plan to alomaker's, if you are not locked in to an existing plan.

I'm not a big phone user, either here or over there, and I don't know how much you guys pay down there, but my plan up here (less than C$40 a month, using my own device) covers me in any destination in Europe I'd be interested in visiting, plus many more places.

I think you can get monthly international plans, which are not cheap in Canada. I don't see the point of a dual sim if are going to be using your US number frequently. You can also use WhatsApp over wifi wherever you are staying, if it's just to touch base at the end of the day.

The dual sim is a function of your phone, not your cellular company.

You could use Whatsapp over WiFi to make and receive calls with your existing phone number provided the other party is using the Whatsapp app also.

BTW, Rick's article on "Using a European Sim" clearly states that a sim card will give you a different number.

If you anticipate heavy phone or data use, consider traveling with a mobile phone equipped with a European SIM card. This gives you a European mobile number and access to cheaper rates than you'd get through your US carrier, even with an international plan.

Good luck!

If you want to keep your same phone number to receive and send calls and texts without costly international plans, consider changing phone company (keep your phone number) to one that allows Wi-Fi calling while overseas. This way, you can make calls just like home while you are connected to Wi-Fi, at the airport, in your hotel or if you need data while travelling for things like Google maps, via a personal hotspot. You can rent or buy a hotspot, put in a data SIM or use an old second phone & data SIM that you can connect to when on the road.

"The info about a dual sim is interesting. I doubt that my cell phone provider company (Credo Mobile) enables this, but I can ask them."

Nothing to do either your phone company- it's all down to what handset you have. It's pretty common - open your SIM tray to see if it has two SIM slots, or check whether it supports physical SIM plus eSIM.

As mentioned, it seems like wifi calling is a good solution. Very old phones (with very old operating systems), may not have wifi calling available. Just poke around in all the menus to find it. Also, some low cost service providers don't provide the wifi calling functionality. You'd have to poke around their website to see if wifi calling is included in the plan you have.

If the OP has a recent enough iPhone with esim or dual sim capability, it's possible that it can magically use data from a second sim or esim and do sort of a "wifi calling" trick. At least, I think so from reading other posts. I know my recent Android phone can't do that.

Like many frequent U.S. travelers here, I'm with Tmobile, and my phone just works everywhere, for no extra money.

What has changed, for Americans anyway, is that dual SIM phones are now commonly available. (They have been available in Europe and Asia much longer)

This means that you can keep your "home" SIM active, and insert a local SIM as well.

When I travel, my phone allows me to choose to have my home SIM used for calls and texts, and the local SIM for data.

I have Google Fi for phone service, so I have worldwide coverage at no additional cost (International calling rates apply). That includes data, but for the volume of data I use while traveling, a local SIM is cheaper and at times better speeds and quality.

The problem that many have, is that their carrier does not provide a cost effective international plan, even without data. In that case, you could still switch between SIMs, local for wandering, Home SIM when you can get WiFi. People would still not be able to get a hold of you most of the time, but at least several times a day you might see messages and texts, and it seems they always call when you are sleeping anyway.

Of course there are still issues of contracts, locked phones, and restrictions that come with getting a "free" phone to navigate, but if you travel often, an operating phone with service is becoming a necessity, so you need to plan for it, even if it means changing carriers.

epltd, at https://www.ricksteves.com/travel-tips/phones-tech/cell-phone-europe, Rick says the following:

Remember that as long as you have a European SIM in your own phone, you won't be reachable at your regular US number.

A few folks have suggested WhatsApp as an alternative. Before you travel, install it on your phone and have your friends/family at home install it as well. Then you will be able to call each other.

Facebook Messenger has similar functionality if you and they are both Facebook users. (By the way, WhatsApp is owned by Facebook, but their two messaging systems are different.)

I think the OP just had to work out what is most cost effective. The last three phones I've had have been dual SIM, so if be surprised if most handsets are not. Having said that, my provider now offers a very cheap roaming plan inclusive of both data and phone calls, so I quite often just use roaming rather than buy a SIM or eSIM.

Good point on whatsapp - here in SE Asia virtually all the calls and messages i send/make and receive (including businesses) are on whatsapp, and it's fairly commonly used in Europe as well (certainly by all my family). I'm in Istanbul at the moment, and my airport driver called me on whatsapp to check my arrival. I find it essential.