Regardless of who actually is this kittys' parents, it seems it would really like to befriend you.

As an animal person, I would take up this offer, but since you have some discomfort with it, perhaps it is better for you to decline.

On the other hand, letting an animal into your life and heart can be very rewarding. Cats are very companiable critters, and can bring a lot of comfort and joy to you.