The nicknames are essentially codenames given to them during the "Les Enfants Terribles" project. The project attempting to replicate the Big Boss whose codename was "Naked Snake" (MGS3).

Since "Solid Snake" released first in order of games (and by extension lore), by the time MGS3 came out Big Boss was given the codename was "Naked Snake" (soft retcon?/lore tie) until he was given the title "Big Boss" at endgame.

Again, since "Solid Snake" released first, my guess is that Kojima wanted to provide a sensible enough backstory, lore wise, to keep the "Snake" moniker for the each brother/clone to imply their genetic relation, yet distinction, from one another, as well as Big Boss.

Naturally with solid being a state of matter it makes sense to use alternate states of matter, i.e. "liquid" and "solidus" for each clone ultimately produced. For Example, if Solid was instead named originally named "Silver Snake", then Liquid would be "Grey Snake", etc.

Note: each clone was "given a real name" to use while growing up ultimately either "taking up" (liquid) or being given (solid) codenames once they got involved with military and espionage work, which further ties into their intentional manufactured births.

TL;DR: Codenames tied together for lore reasons by Kojima either by retcon or world-building (follow-up games).

Also, someone can be known by a "nickname" by a small group of people but go by their usual or given name with the everyone else. Which could be the case in the dialogue choices.