Cats rubbing their face on something leave their scent behind / mark their territory, so yes, in a way they’re claiming ownership. However, your cat isn’t saying “this item is mine now” with this message, but rather likes to leave its own scent on items that also smell like you, as they associate it with something comforting and safe. It is more like a “this thing smells like my human and I want other cats to know that we belong together” scenario.

As for why the TV remote of all things: you’re holding that thing in your hand, and your palms have a lot of special sweat glands that do not react to heat but rather emotions. Humans get sweaty palms when stressed or exited and your cat is basically able to “read” these emotions in items that you held in your hands. If you’re relaxed and comfortable while switching through the channels, the TV remote will basically smell like “my favorite human is happy” to your cat. ;)