Time Warner Cable was spun out from Time Warner, Inc. seven years ago, and they no longer have any relationship with each other. Time Warner Cable was permitted to continue using the name under license, though they are now trying to rebrand themselves as "Spectrum" in markets where they already had a presence (such as New York City).
AT&T is reportedly talking about buying Time Warner, Inc, which is a media conglomerate that owns either part or all of brands like Warner Bros, CNN, DC Comics, and Hulu.