The anointing is the manifestation of the power of God through the life of a person. You can have the holy spirit wen born again but you need the spirit upon you and that is the holy spirit which also acts as the anointing.
After one is baptised, the spirit upon them doesn’t increase but the power of the spirit does.Jesus felt power go out of him when the woman with the issue of blood touched him, Jesus was anointed with power.
When the anointing is present, evil things are lifted. because of the anointing,the unbelieving king Pharoah recognized the anointing on Joseph when he interpreted his dream to him.
Scriptures to assist you understand the anointing better;
1st Samuel 10:1 & 9
1st sam 16:13