Yes, it’s absolutely possible to learn weight lifting on your own, however, it’s not something most people would recommend.
Watching workout videos and studying correct form and technique will enable you to get started on basic weight lifting techniques and programs and can be a way to get started for people who are nervous or shy.
However learning the correct technique, which weight lifters often refer to as form, is very important as incorrect form can lead to lasting damage to your joints, muscles, and posture which can be difficult to correct.
Ideally training with an experienced trainer at the beginning of your journey will help you to avoid bad form and bad habits and get you started on the right track. Once you’re more experienced and confident going it alone is a much more reasonable option.
Just don’t forget that you should still use a spotter for those big lifts and compound exercises.