“Practise the piano” vs. “practise medicine”

The short story is: the "apparently opposite" meanings are in reality not opposite at all; they are merely applied to different spheres.

Dictionary.com on practice:

Origin:
1375–1425; (v.) late Middle English practisen, practizen (< Middle French pra ( c ) tiser ) < Medieval Latin prāctizāre, alteration of prācticāre, derivative of prāctica practical work < Greek prāktikḗ noun use of feminine of prāktikós practic; see -ize; (noun) late Middle English, derivative of the v.

I put the original meaning practical work in bold. From here, it is easy to derive the two current meanings: practicing the piano is practical work if you want to get better at it; practicing medicine is practical work if you are good at it and want to keep a job. They're just two senses of the same thing. It doesn't require a large stretch of imagination to go from practical work to either current meaning.

There might have been more to it, but if you take etymonline:

practice early 15c., "to perform repeatedly to acquire skill;" mid-15c., "to perform, to work at, exercise," from O.Fr. practiser "to practice," from M.L. practicare "to do, perform, practice," from L.L. practicus "practical," from Gk. praktikos "practical." The noun is from early 15c., originally as practise, from O.Fr. pratiser, from M.L. practicare. Also as practik, which survived in parallel into 19c. Practiced "expert" is from 1560s; practicing (adj.) is recorded from 1620s in reference to professions, from 1906 in reference to religions.

the meanings are not really opposite. Original meaning "to perform repeatedly to acquire skill" keeps the meaning of "to perform repeatedly", but changes it slightly to "performed repeatedly and became an expert." This change is explainable by common use of the word.