Xiaomi Mi Band 2 cannot be paired with App Mi Fit in Android 8.1.0
So in the end, the solution was incredibly easy. At the time I was trying to pair it, it never occurred to me to try anything other than the official app. But since then, I've found that the official app may not always be the best. And here it was the same - it was enough to use another application - Mi Band App for HRX, 2 and Mi Band 3.
I haven't tested it yet, but the pairing itself worked, so hopefully there won't be another problem.