Can meditation strengthen the sense of self?
Just do what you're doing. When you notice your self, don't fight it. Your sense of self is there for a purpose. The purpose is to find out the cause. Once the cause of self is found, the cause is removed. When the cause is removed, self ceases.

When meditation is done gently, spontaneously, peacefully, one becomes absorbed in meditation. When one is absorbed in meditation, thinking ceases. When thinking ceases, self ceases.