This is another phrase where a single word has confused people – 'fit' in the context of this saying does not mean 'healthy', which is a 19th century definition. Its original meaning was 'suitable' – and it is still used in that context in the saying 'fit for a king'.
As fit as a fiddle means 'as appropriate as can be' – not “in excellent health”. The first use of the phrase was in the 16th century and it was originally 'as right as a fiddle'.