Woods is sponsored by Rolex. Like in F1, he slapped it on at the end to meet his commitments to Rolex, the same as the F1’s drivers do for their own sponsors.
I actually bought a Rolex Yachtmaster which was gifted to a Spanish golfer whose name I cannot remember and it had his name on the papers. They often give out watches free to a number of the players at tournaments as they sponsor the timing. Most of these watches then get sold as they just don’t want them.