That is a very sweet blade.
Unfortunately the scales (handles) look to be cactus.
If the original scales are celluloid lifting the inlay is a pretty simple operation, but from the photos it looks like the scales aren't celluloid.

The leather sharpening sheath is a strop, but what you have is a very rare and unusual example of one. It also looks to be specifically made for your razor which isn't something I have seen before.

If it were me I would be looking at giving the whole kit a complete re-furb. replace the leather on the strop and sand & polish the handle, restore the blade and a new set of scales. My choice would be to try to replicate the original scales, but it's your razor.