The short answer to this is "no", you can't. It is not customizable. Ok, that depends also on whether a (or the lead) developer wants to implement this feature (user-definable sounds) in the near future or allows you to directly add/modify the public sound sets, though I wouldn't count too much on it.
Meanwhile you can try to install a user script to set the sounds for move, capture, check, victory, defeat, draw, low on time,... to sound files you like. For this you will need to first install a user script manager (Greasemonkey, Tampermonkey, or Violentmonkey depending on which browser you use) and then find or write (if you know JavaScript) your desired user script.
There are many websites available to search for your favourite scripts. One of the best starting points is Greasy Fork, a site for user scripts. Browse this site to find a sound replacement script for Lichess you want to try such as:
Lichess Old Sound
greasyfork.org/en/scripts/8387-lichess-old-soundLichess ChessCubePack
greasyfork.org/en/scripts/8466-lichess-chesscubepackLichess ChessComPack
greasyfork.org/en/scripts/8474-lichess-chesscompackLichess Fritz Sound Pack Sound replacement(only)
greasyfork.org/en/scripts/8550-lichess-fritz-sound-pack Lichess Blitzin Pack Sound of Blitzin
greasyfork.org/en/scripts/8502-lichess-blitzin-packLichess ChessBase-ChessMaster Mix Pack
greasyfork.org/en/scripts/8487-lichess-chessbase-chessmaster-mix-packFinally, click the install button on the script's page and confirm it with the script manager. When you visit Lichess again the script should replace the sounds without user intervention.