I have not heard of liquid dish soap damaging paint other than removing wax.
Powdered soap might be a problem.
Many autobody people use household laundry detergent(also job specific products) to remove waxes & silicone from new/old plastic parts prior to repair,sanding or painting. Dishsoap is milder but will remove wax. I can also attest to the fact that it dries out your skin from having bathed(me & everybody else too) in it to get off carbon black after a tire plant shift.
Liquid car wash soap isn't going to hurt your vinyl seats. They are an outdoor part. I don't see why anyone would put leather goop on them either as has a different purpose. Keeping the sun/UV off them is the best favor you can do them.