Matchlines are typically a feature found on plans, not maps. I'm not aware of any tool to directly implement them in ArcGIS. It's possible there are such tools in some of the mapbook extensions available out there. However there is a workaround that can roughly do it.
First, you need to create a new feature class and actually draw in the matchlines where you want them. Sometimes matchlines are shown in different places on their matching pages rather than right at the 'official split' - that won't be possible here. You'll need to create single lines. You'll then symbolize that layer in your map in some obvious way (overly thick and dashed for example).
Once you have the lines, you'll need to potentially modify your reference feature grid. If it doesn't already have them, you need adjacent neighbor fields. There's a tool for this that would work in some cases (Calculate Adjacent Fields), otherwise you just manually add an attribute and populate it yourself (which can also help control what your match line text says).
With that, you can insert a Dynamic Text element into the layout near the edges of the page. Your map view will rotate as you go from page to page, and it's possible the exact location of the matchlines may move as well and their rotation definitely will. However these text elements won't, so you need to locate them somewhere that will generally be near where the matchlines will appear.
It's not pretty or ideal, and is basically just taking adjacent page labels out of the map collar and moving them over the map itself. Related: How to label adjacent pages in data driven pages
With that network of lines, there's no way for ArcGIS to figure out which page you'd like to be "first". The strip map tool is meant to create a mapbook along one line feature, it doesn't do so well with more than one, especially if they crisscross each other like yours do.
However, it does look like it covered all of your lines with mapbook pages, and they're oriented more or less correctly. In your situation, I'd just edit the index layer and re-number the pages however you want them to be ordered. I would also remove some of the pages, when you have overlapping lines and you use the Strip Map tool, you can get unnecessary pages where your lines meet, because the tool is generating a complete map set for each line feature individually. Pages 24 and 48 in your screenshot, for example, seem to be redundant.
If you want to be able to generate the pages automatically, you will probably have better luck using the Grid Index Features tool instead.
Best Answer
The tool Strip Map Index Features (Cartography) does exactly what I was looking for because it: