It's possible...thinner the material the easier it is...
It's super easy to remove it from the fabric pot... if it's ready for a transplant, the roots should hold the medium together along with some minimal amount of moisture will prevent it from falling apart. If roots have grown into the fabric, run a garden trowel along the outside until you free it up... can also fold down the sides of the fabric and slowly slip it out that way too... many ways to skin this cat.
Trying to transplant a tiny plant in a large pot may not be as easy.. the roots won't hold the medium together... if that happens, use smaller pots that fit your timeline, next time. or, let them properly grow into that pot, first, then transplant.
Yes, the roots probably would grow through it, but it'll certainly slow it down, too. It takes minimal effort to pull it out, so might as well do it before putting in the earth.
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Those look like velcro pots in the 2nd picture.. Are you goofing around?