I think the important point is that Hungarian does not have a separate word for "son". You can not express it without the possessed ending. It is always expressed as "(somebody's) son". That is why it looks as though the word for "son" is "fia".

Unless we want to use "fiúgyermek" or something like that.

The difference between "my son" and "my boy" is expressed with the different possessed endings. The regular ending ("fiúm") is "my boy" - or, idiomatically: "my boyfriend" - and the irregular ending ("fiam") is "my son".