As user6726 mentions, acrophony (and the related rebus principle) appear commonly when developing writing systems in the first place. If the word for "mouth" is pronounced
/ka/, then people might use a drawing of a mouth to mean
/ka/ (rebus), or simply
/k/ (acrophony). On the flipside, letters may be named after words starting with that sound, as a sort of mnemonic; this is also called acrophony.
Beyond that, though, the question of "what do the components of this writing system mean to a particular culture" tends to be more anthropological than linguistic. There's a bit of linguistic work on the topic of "phonesthetics", the idea that certain sounds are fundamentally linked to certain broad categories in the mind, but the majority of words and word roots in natural languages are arbitrary—empirically speaking, there's no evidence of a fundamental connection between the sounds
/t/, and a cat. It's just a historical coincidence that we use that combination of sounds to refer to that creature, because our ancestors used a similar combination of sounds, and their ancestors used a similar combination of sounds, and so on.
There are also various philosophical and religious traditions that assign meaning to different components of a writing system, such as gematria. These are, likewise, not generally considered to be a part of linguistics. The point of these systems isn't generally to be scientific or falsifiable, so the tools of linguistics aren't very useful for analyzing them.