Yes, there is. It allows fans to enjoy the melody without the vocals, which can be great for various uses like background music while studying or just relaxing.
One common element is a shared activity. For instance, if it's an art - related instrumental love story, painting together could be that element. Another could be a specific location. Just like in the example of the library before, a place can be a common element.
An 'instrumental love story' could be a love story where music or some other form of art or tool (instrument in a broad sense) plays a significant role. For example, two musicians might fall in love while playing music together, and the music they create becomes an integral part of their relationship.
Austin and Ally's love story is really sweet. They start as friends who are very different. Austin is the confident performer, while Ally is the shy songwriter. As they work together on music, their bond grows. There are many cute moments where you can see the feelings developing between them, like when they share glances during performances or when they write songs together that seem to be about their unspoken feelings for each other.
Taylor Swift herself might have an official instrumental version. Since she is the original artist of 'Love Story', her version would be a great choice.
You can try YouTube. There are many channels dedicated to uploading instrumental music, and you're likely to find great instrumental versions of 'Love Story' there.