How do chess players keep track of their past games for training? Special motifs, instructive mistakes, comments on positions and so on? Writing down moves on paper seems anachronistic to me, especially when many online games are played. It also doesn't allow for quick computer analysis or searchability.
There's a plethora of chess software that support saving, querying and updating games played. From the commercial ChessBase to open source equivalents like SCID on PC, ChessX and Tarrasch. They all support database facilities that allow games to be added, searched by tournament, player, position, pawn structure, opening, in some cases even by tactical motif and strategic themes. And with plugins for various chess engines, analysed and blunder-checked.
For a player's own games, if the tournament is run by a good set of arbiters, then the games will make their way to The Week In Chess weekly chess databases that a player can download and open in their chess database software. Players can also enter their own games into their personal chess databases that can be queried, searched, analysed and dissected in the very same fashion.
For online chess playing sites, player's games can be downloaded individually or as a collection and opened in their chess software.
These chess databases and their software are flexible enough to create subsets or aggregates of other chess databases, so a player, or trainer, can typically create their own personal/private databases formed of games or positions that meet their training needs.