Does trypsin strip flask coating?
Generally the coating is of poly D-Lysine. Proteases act only on L-amino acids. So, I guess nothing should happen to the usual coated flasks. I have reused the flask even after 2 rounds of trypsinization, and the cells seemed to be fine.

In your case, the matrix is made of proteins isolated from animal sources. They are susceptible to proteolysis. You are right to assume that these flasks should not be reused. If you have a shortage of flasks then you can buy uncoated petriplates (which are cheaper) and coat them yourself.