But if we say "קוראים לי דני"...
....then we do not say that "someone 'reads' me Dani," right? It would make no sense! We mean "my name is Dani"!
But why do we use the word "קוראים", then? The answer is simple - because what we actually say is "(they) call me Dani"! (omitting the word "they.")
How did "read" turn into "call"?!Well, if we think about it, "reading", if we read out loud, is not that different from "calling," like "calling out sounds"! :) But - there is one significant difference!
If we say "he calls his aunt 'coach potato'' we say
"הוא קורא לדודה שלו 'בטטת קורסה'", which actually means -
"he calls to his aunt 'coach potato' "
Without this tiny "le-" it would mean that "he reads his aunt" - and totally wouldn't make sense. :)
Another example: - 'אני קורא לו 'חבר - I call (TO) him a friend.