No, it doesn't. The near-60% positive response from the AP poll you're linking is in response to this question:
"0. Should couples of the same sex be entitled to the same government benefits as married couples
of the opposite sex, or should the government distinguish between them?"
That's not the same thing as saying one supports equal marriage; that's perhaps an argument for saying that folks aren't opposed to what gay people are forced to deal with right now - i.e. a hodgepodge of legal arrangements, powers-of-attorney and so on that come with marriage rights by default.
Answering in the affirmative to the above question doesn't actually move the needle on support for equal marriage; it merely means that 57 and 58% majorities during two rounds of polling (less in subsequent rounds, at 54 and 51%, with a 4 point margin) are comfortable with the easy and obvious claim that if I'm gay and in a relationship I and my parner should perhaps have things like hospital visitation or shouldn't be denied access to publicly funded services.
That's not actually support for equal marriage rights. A better look at what your preferred poll is saying is in this question:
"Should the government give legal recognition to marriages between couples of the same sex, or
Here, we've got three rounds of polling with Yes responses at 53%, 52% and 46% with a four point margin of error overall.
It would certainly be nice, for me, if even your poll suggested what you're claiming it does, but it doesn't do that.