Also, the eye only saw Frodo because he was in the Seat of Seeing in Amon Hen, which is enchanted to act as a sort of observatory. While long abandoned, the enchantment was still active. It allowed Frodo to see far into the distance, possibly enhanced by the ring, it also allowed Sauron to see him back, similar to the Palantir.
Sauron wasn’t “back” yet when Bilbo had the ring
Also, the eye only saw Frodo because he was in the Seat of Seeing in Amon Hen, which is enchanted to act as a sort of observatory. While long abandoned, the enchantment was still active. It allowed Frodo to see far into the distance, possibly enhanced by the ring, it also allowed Sauron to see him back, similar to the Palantir.