I haven’t found a service that reliably works will all banks/services across the world. The problem is that many banks refuse to work with any virtual numbers, they require you to actually have a physical SIM card.
So the most reliable (and still relatively easy) solution is to get a cheap “dumb phone” where you put the SIM you use for 2FA and bring it with you wherever you go.
Just make sure that the phone supports phone networks across the world (look at quad band phones, they typically do) and just in case you need to pick up another SIM card to work with a specific bank/country later, I’d also recommend getting a dual sim phone.
The SIM card/subscription itself should support international roaming and be able to receive SMS for free—which is the case for most non-North American providers I’ve seen.
If it makes most sense for you to keep a Danish SIM card, you could move your number to Oister.dk—they have a plan where you only pay 19 DKK per month.
In other countries it might be easier to find pre-paid SIMs which can be topped up online every now or then to keep the number alive. This is often cheaper. For example, to keep my Norwegian number alive I only have to top up 25 NOK (~€2.5) every 15 months.
If you’d rather not bring an extra phone with you, you could use an Android smartphone instead (you can find these for pretty cheap) that you leave connected to WiFi at your home base (if you have one) or with friends or family. Then you install an app on the phone that will forward any SMS that you receive to your email.