Synthetic saddles come in several different materials. Cordura, for instance. There wouldn't be any benefit to putting oil on Cordura, But you could probably wash it with mild soap, water and a soft scrub brush. Getting the dirt and sweat out would make it softer. Then you could apply a waterproofing spray, like Nikwax or Scotchbrite, or Camp Dry. That's what I've done with my Cordura saddle.
A Wintec would be made of a non-absorbing, suede like material that would probably be really damaged by oil. It wouldn't absorb the oil and would surely mess with the "sticky" quality of the material that is great for keeping you in the seat. Oil would surely ruin the "nap". Wintec makes a wash that you can use to clean the dirt off, and there are vinyl cleaners and protectants that might be good on the smooth under-surfaces. Raggtop is a brand of cleaner/conditioner that is not slippery and is made for auto Convertible tops, so has lots of UV protection in it. My Wintec is 18 years old, gets brushed occasionally to get rid of the dust, maybe wiped with a damp rag. It is virtually as good as new.
Even some leathers, like the seats of many western saddles, will reject oils. My Big Horn Cordura saddle, has a seat made of virtually impermeable leather, while the surrounding leather drinks conditioner by the gallon.
Other brands, other materials. But most synthetic materials are non-absorbent and would typically not absorb oils.