You can but how old is that tape, what is underneath the tape, how old is the housing and what shape is it in?
You sweat and it gets soaked up by the tape and it just sits against your bars doing who knows what and yeah it might be a little gross in the end it is a little corrosive and can wreak havoc on bars and you wouldn't know till you remove the tape. However in the end this gives you a great opportunity to upgrade all of your cables and housing and get some higher quality stuff for not much money and get better braking and shifting without spending a ton. Plus you can upgrade your tape as well if you want something more or less cushioned or maybe a different feel entirely.

Yes you physically can just replace a single cable but chances are that is the same old stuff that came with the bike and the bike is probably at least a few years old and probably more so doing some good maintenance on it and a little overhauling would be a net good and help the bike last longer but more importantly work better.