SMS is interoperable between every phone and phone company on the planet.
I've read some porting horror stories online over the years. Yours is probably something like the 6th or 7th. I assume things do sometimes break, but it doesn't seem to be particularly frequent, considering millions of people switch carriers every year.
I will say this though, switching mobile goes rather well almost all of the time. Porting numbers on fixed telephone lines is much more apt to fail.
When I (UK) switched in the late 2000s, it took about an hour.