I have pondered this same question extensively, unfortunately I cannot give you a straight answer on specifics of how rubber handles are actually made. I am around ten books into studying plastics and elastomers so I have some general knowledge but no insight into proprietary process.
Here is what I have so far. Natural material is generally latex, gutta-percha would be superior for handles but very uncommon now. Synthetics are very wide range like neoprene, butyl, EPDM, Hypalon, Viton, buna, silicone rubber, and nitrile for just tip of iceberg. Elastomers are also commonly cross-bonded with plastics for various properties needed. If a particular handle names the rubber used this may help you find out specifics.
Natural rubber latex can be purchased at a large craft store as mold-builder. Paint it over objects you wish to reproduce and then cast into it, I use this process to replicate models I carved many years ago. I do not know if this compound would set up easily in large quantity without heat etc. But the quality of the latex is very good, and latex can be hardened through heat, vulcanization.
Off the shelf products to experiment with are Lexel, plasti-dip, flex-seal, and truck bed liners. A call to a company that provides elastomers may yield great results, right now the limit of my usage is covering cast aluminum handles with coating. Butyl rubber may also be available in tube form from a good supplier. Some compounds can be hardened with heat so although standard silicon may be too weak vulcanization may change this.