Philippians 3:5 tells us saint Paul's family heritage not his religion and Acts 32:6 also tells his family heritage but not his religion. Surely there is a difference between the faith of Christians and the religion of the Pharisees who were scrupulous about the Laws of cleansing and separation but who denied Christ and participated in the plans to kill him and in his death?
There are no born Christians just like there are no born Muslims or born Pharisees. There are even no born Jews from Yeshua's perspective, He addressed this when some Jews said they were descended from Abraham and were Abraham's children. He acknowledged that they were physically Abraham's descendants but not Abraham's children. Further He described them as the Devil's children because of their actions and their intent towards Him. This is recorded in John 8.
Saul/Paul asserts his Pharisee status in those 2 passages to either assert some sort of religious authority and/or to do the same when he is comparing himself to what he calls the "super apostles". His statements in context are not just to give his "heritage" but to command attention to those whom he is speaking because of it. It is part of his double speak: "Listen to me - I am a Pharisee, these are my qualifications!" then "but I count it all as garbage for the sake of knowing Christ...".
Furthermore, simply mentioning the name "Jesus Christ" or referring to the Messiah or any other title by which He is known does not mean acknowledging him - OTHER THAN IN NAME. Most Christians do not accept the Gospel of Thomas, or the Book of Mormon as inspired works called "Scripture" and yet these are works that frequently reference a person who is called Christ. The reason I point this out is to highlight what you say about "denying Christ". People think of this phrase as simply unbelief or denial of the person of Yeshua, but this is not necessarily so.
The term "Anti Christ" may mean "against Christ" - but it does not describe the method for the attack. According to Saul/Paul's own words he was an enemy of early believers and murdered them. That is unquestionably anti - Christ. But after his so called conversion he is still anti-christ. Not in the sense of murdering believers anymore - but in twisting the words of Scripture (OT included) and of Christ and replacing the Teachings of Yeshua with teachings that contradict Yeshua. Saul/Paul does this masterfully, he mixes in truth with error, and with nonsense. It is near impossible to spot if the blinders of "the bible is the complete Word of God" are on - readers won't look up his quotes or check what he says against the teachings of Yeshua nor what he quotes from the OT. In this way his denial of Christ is done by subtlety, not overtly, and with great deference to the name "Jesus Christ" and tales of personal sufferings and faith that serve to cover them up and convince everyone he is a super apostle, when he is in fact a major anti-Christ.
