There are some fabric pots that are designed to do this, mainly meant for seedlings. They are a texture like dryer sheets and roots can pass through them. These work great as a starter pot, and I've been using them for a few grows. I have also had roots grow right out and through 5 gallon fabric pots due to my frequent fertigations, and they would have spread like crazy if they were not air pruned. But I have not tried what you are describing with thicker fabric pots personally, even though it should be totally possible. There may be some fabric pots that would restrict this kind of growth so I don't want to say it's possible in all cases.
It would probably grow alright if directly transplanted in the pot, but there are some other possible issues that could arise. The extra layer of fabric could cause increased water retention in the original pot, and inhibit oxygenation of the roots for example. But if you controlled for this with a well aerated grow medium and good watering practices you could probably manage just fine!
If you decide to transplant out of your fabric pot, you can cut the pot open if you are very concerned about getting it out. Either way you would be losing the first pot as the roots growing through would wreck it, so that could be an alternative. I find it also helps to make sure the pot is very well populated with roots and has a pretty compact root ball. Then you can let it dry out just a little bit, give the underneath a few quick firm pushes, and start turning the pot inside out. A blunt knife or something similar can be used the pry the roots away from the fabric if you find they are a bit stuck. Don't worry if you lose small roots right at the edge, what really matters is that the root ball stays intact.
So to sum it all up in my opinion/experience, it's likely possible, and you can do it if you are curious about the results. BUT there are the risks I described which is probably why you don't see it as a common practice for thicker textured fabric grow pots. All the best and happy growing! 👍🌱