https://arstechnica.com/science/2016/02/the-audacious-rescue...

Basically yeah, it could have been done, but only by pulling out one of the other shuttles out of active maintanance, skipping a whole bunch of repairs and necessary checks, launching in record time, and actually managing a situation where there are two shuttles in orbit, something NASA wasn't exactly prepared for at the time.

And the biggest question of all - what if the second shuttle becomes damaged in the exact same way on the way up?

That's not to say that NASA shouldn't have at least attempted a rescue. Knowing anything about the way US approaches situations like this, it would have been a "no expenses spared" effort and they would at least try. But unfortunately it's extremely likely that the second shuttle would have also been lost on the way, with crucial maintenance and checks skipped just to make it space worthy in time.