General: This is the most common synonym, meaning "relating to or applicable to all or many kinds or cases."
Universal: This implies something that applies everywhere or to everyone.
Common: This suggests something that is ordinary or widespread.
Standard: This emphasizes something that is widely accepted or used.
Non-specific: This highlights the lack of detail or individuality.
Typical: This suggests something that is representative of a group or category.
The best synonym for "generic" will depend on the specific situation and what you are trying to convey.