If I understand this correctly, you want the polygons to have an attibute listing of all the roads in that polygon.
Clip is likely not the tool you want to use, although it would be possible to use it as part of the process.
Instead, try intersecting the lines and polygons (Intersect tool in the analysis > overlay toolbox)
This will create a line feature class with the lines broken at the polygon edges and the attributes of the polygons added to each line feature. This may be all you actually want or need.
getting the (list of potentially many) roads through each area attached to each polygon takes more work.
A Spatial Join (with one-to-many) of the intersect output roads to the polygons is one step towards doing that.
