Yes, it was built so UConn football could use it. However, it was also built so other organizations could use it too. Ownership of the stadium is important because that dictates who has control over future decisions for the stadium. The state government could decide to renovate or expand Pratt & Whitney Stadium and it wouldn't need UConn's approval. However, if UConn wishes to renovate or expand the stadium, then it would need to get the state's approval because at that point the state (via whomever the state placed in charge of managing the stadium) would have to handle all contract negotiates for whatever work that needs to be done on it.

The situation in Connecticut is a lot different than in Virginia. If VT wishes to renovate Lane Stadium (or any of the other athletic facilities), then it doesn't need to get approval from the state because taxpayer dollars are not being used to fund the project. Likewise the state government can't make any decisions on VT's athletic facilities without at least getting input from VT first. Even then the state may still differ all decisions to VT since the state of VA decided that no taxpayer dollar shall be used to fund collegiate athletic capital projects.