I think coins were collected fairly soon after the concept of a coinage was introduced.
The evidence is certainly there for numismatic hoarding by the Roman period, because the Bolsena hoard, sold as part of the Sarti sale in 1906, contained practically as struck sestertii covering a period of 200 years or so. There is no way that high grade sestertii would survive in circulation for so long, leaving a numismatic collection as the only sensible alternative.
The artistry employed in engraving coins would be as recognisable as that seen on jewellery or other artefacts, so there is no obvious reason for not collecting them.