A server I manage has Oracle Client 19.16 with Java JRE 1.8_331 installed. Can I just copy a newer version of the Java JRE, like 1.8_351 over the old one without having any negative effect on Oracle Client? I'm limited to only using the 19.16 patch, but an older version of Java is being flagged when the workstation is scanned.
Generally speaking, no you can't just do that. You would download and apply a quarterly patch from Oracle for that purpose, or a newer version of the client. Don't just replace embedded functionality like that unless you've got specific documentation or instructions from Oracle Support that validate the procedure and explain how to do it correctly.