Have is rarely used in passive constructions, even when it is used as a dynamic (versus a stative) verb.
Here are two acceptable uses:
1 A good time was had by all.
This expression (and ones based on it) is quaint, hackneyed and stale.
More germane to your question:
2 I was had by the insurance company.
Here was had means was cheated or was swindled.
See Oxford definition 2.8 and be had at the free dictionary.
Now, let's look at a particular active usage that is ungrammatical in the passive:
Had can mean gave birth (simple past). A common but somewhat informal use is
My mother had me in 1999 when she was still young.
Had here means gave birth to.
However, to use this meaning in the passive, as in
*>I was had by my mother in 1999 when she was still young.
is ungrammatical. In fact, the natural way to interpret this sentence would be to give it the meaning of to be cheated.
Which is why I originally jumped to the conclusion that was had in the lyric you ask about means to be cheated.
However, the songwriter has been clever and written a lyric that is both grammatical and ungrammatical at the same time. The grammaticality of its use as I was cheated/swindled "allows" the listener to process the ungrammatical usage as I was given birth to as making sense in the context of the singer's parenting. It's a startling usage.
If one were to distill or separate the two uses, the lyric would mean something like:
Somehow I was cheated when I was given birth to by my parents.
But again, I was had as in I was given birth to is strictly ungrammatical in normal usage.
Acknowledgement to the comment
by MarkHubbard and the answer
by Codeswitcher, which have allowed me to update and improve my answer...