How To Create Hair Cards with Gemoetry Nodes
... and How To Convert a UV Map Calculated by Geometry Nodes To A Real UV Map
To make your example work you need to change two things:
- make the length and width attributes a vector attribute (my uv map) because only vectors can be converted into UV maps.
- convert the vector attribute into a UV after the hair cards have been converted to a mesh. And use the new UV map in the shader, of course.
1. Creating the Attribute "my uv map"
This is done quickly by combining the formerly length and width attributes to a vector with the Combine XYZ node. Set Z to 0. Then expose the vector via Geometry Nodes group output (or use a Store Named Attribute node.)

In the Shader Editor you can use this vector attribute as a UV map. Here are the GN hair cards with a material:

2. Convert the GN Hair Cards For Export
2.1 Convert the Hair Cards to a Mesh
Converting the Geometry Nodes hair cards to a mesh is a bit weird in Blender 3.3 and might change in the future. Right now, the menu Object > Convert > Mesh does nothing. And when you try to apply the Geometry Nodes modifier of the hair curves object (Curves) you just get an error.
The workaround is
- add a dummy object: A plane named Hair Cards to Export and place it where the head is (Suzanne)
- add a Geometry Nodes modifier that just exports the geometry of the Curves object. Then apply this modifier and you have the mesh with the hair cards.

2.2 Create the UV Map
For Blender 3.5+ (quote from the documentation):
In order for Blender to recognize the created attribute as a UV map, it must be created with the Store Named Attribute Node on the Face Corner domain with the 2D Vector data type. This is necessary because there is no 2D Vector socket type.
See example here: How to convert UVMap attribute to UV map in blender 3.6
For previous versions (Blender 3.3 and 3.4):
Finally, convert the attribute my uv map into a UV map. This is done in the Object Data Properties panel. Select UV Map for Mode and press OK.

Last but not least, select the material and use the newly created UV map. Here is the material with the UV map:

Related question: Geometry Nodes, Hair, and UV Unwrap: How to make UV alpha cards along a strand for hair cards with multiple strand images.