Oily residue on old toy rubbery tyres
The &quot;rubber&quot;, or elastomer, likely has plasticizers to soften the material and make it more elastic and less brittle. These often migrate to the surface as parts age. Some of these plasticizers for elastomers include:

Sebacates, similar to animal fats
Adipates, based on adipic acid, which is found in beets
Phthalates, esters of phthalic anhydride
Some are innocuous, but others, such as bisphenol A (BPA) may have long-term toxic effects. Washing the parts, as you have done, should remove them from the surface. You might need to repeat washing, using different detergents.

Another possibility is depolymerization, actual breakdown of the plastic. At room temperature, this process is slow, though it could contribute to that sticky surface layer.
As for safety, you might check on rules at the time and place the toys were made. Modern toys are made with safety in mind, but plastics have had a bad rap in the past, such as causing liver damage to workers in vinyl production. Caveat emptor.