The charmed condition provides the following benefits to the charmer:
A charmed creature can’t attack the charmer or target the charmer with harmful abilities or magical effects.
The charmer has advantage on any ability check to interact socially with the creature.
As you’ve stated, nowhere in the spell’s description does it state that the target is charmed by you.
As such we can confidently state that you are not the charmer, meaning that none of the benefits of the charmed condition apply to you.
I suspect that you may have additional questions related to this, so here is some further analysis.
Who is the charmer anyways?
Taking an English idiomatic interpretation of the spell’s description, the following phrase gives the implication that the spell itself is the charmer.
While charmed by this spell…
There are no rules for being charmed by a spell so the implications of this fall under DM purview.
Does it matter?
It might. The spell’s description states that “while charmed by this spell, the creature is incapacitated and has a speed of 0.”
The incapacitated condition has the following effects:
An incapacitated creature can’t take actions or reactions.
While an incapacitated creature cannot take the Attack action, there are no rules which state that social interaction requires an action, and, unlike the stunned condition, which states that affected creatures “can speak only falteringly”, the incapacitated condition does not restrict a creature’s ability to communicate.
This is significant because it implies that the caster of Hypnotic Patter does not have advantage on social interactions with an affected creature, despite the creature being charmed.