Yes, they had to adopt him (even though they believed he was the real Paul) because blood tests were inconclusive and DNA testing was not available back then. Also, back then, ear shape was considered a good identifying feature, and the ears of “real baby Paul” and “Found Paul” were thought to be a very close match.
However, because it couldn’t be proven, the state did insist that the Fronczaks adopt Paul through the usual channels.
If I remember correctly, Found Paul was clued in to the entire story around age 10 when he stumbled across his adoption papers.