Many musicians can tell intervals, i.e. guess what a note is once they have been told the name of another note.  That's relative pitch.

A smaller group can often start some songs on the correct note.  This happens with songs they hear or practice often typically.  And they will occasionally make mistakes.  (I am in that category: I will get the start note of some songs right more than half of the time; when I am wrong, I am usually off by a tone or so, and sometimes I am _completely_ off.)

Very few can systematically start a song on the correct note, or, given a note, tell instantly which song starts with that note.  In my opinion, the last step of telling the name of a note as opposed to naming correctly, effortlessly, which song starts with that note is not a requirement to perfect pitch: knowing the note name is a detail that you can master easily if you already have associations to each note without thinking about it, and that last skill isn't related to your ear per say.

For example, if you can almost always sing a C because your favorite song starts with a C and from there you derive the relative interval to, say, an A, and can then sing that A pretty accurately, you have a good ear, but not perfect pitch.

In your case, you say you associate "certain" notes to its song.  If you never get that wrong, you are closer than most to perfect pitch.  But until you have an association of each note sound to a correct representation of that note, be it a song name or note name, you would not have what is called perfect pitch as far as goes my understanding of perfect pitch.

People with perfect pitch will be able to name quickly all the notes of a melody they hear effortlessly.  They don't have to think about intervals between any of these notes.  You may still have a way to go for that; most are :-)

By the way, there are some methods to help people acquire perfect pitch, and they do start exactly with what you do: being able to associate sounds to concepts: songs you know, colors, textures, etc. so all of this is a continuous spectrum...  A friend of mine got very near perfect pitch with that approach so people have a margin for improvement: it doesn't have to be a gift though it can be just that.