Is it dangerous to use different sizes on the two tires?

A few days ago, a nail pierced my rear tire at an angle, so I bought a new tire and inner tube.

The problem is, I didn't notice until today—when I went to install it—that the tire they gave me was a 2.125, whereas I had asked for a 2.25. Since several days have passed, I doubt the shop will exchange it. According to AI (ChatGPT and Gemini), it isn't dangerous; in fact, they suggest it's a good idea to have a wider front tire and a narrower rear tire, provided the tread pattern is the same on both wheels—but I don't fully trust AI.

I want to ask if it is dangerous to ride a bike with different tire sizes: a 26x2.25 on the front (although the tire says that, to the naked eye it looks like a 26x2.35) and a 26x1.25 on the rear.