Re: What does Catholic doctrine say about the mark of the beast and what it is?
There is no official teaching on the mark of the beast - 666 (or 616 depending on theory). It was mentioned in The Book of Revelation 13:17-18 and associated with anti-Christ and Satan and Church Persecution.
At the time the apostle John was writing The Church was under heavy persecution by the Romans. Christians had to meet in secret and communicate with code words and symbols to avoid persecution. Many believe that number "666" refers to a Roman Emperor such as Nero (whose name, written in Aramaic can be valued at 666, using the Hebrew numerology of gematria where each Hebrew letter corresponds to a number). But even here there is a dispute since at least one NT manuscript suggests that the original number was actually 616 -- and why some contend that it refers to the Roman Emperor Caligula or somone else.
Irenaeus (second century ECF) reported several scribal errors of the number lending to the theory that 616 might be the real number.
In genesis Man was created on day 6 - the same day that the animals were created and since God rested and completed His work on day 7, 6 is a number often associated with imperfection since man's fall from grace through temptation by Satan brought such suffering to all of Creation. So there is potential for multi-valued and nested meaning.
I note the irony too in the Catholic Catechism paragraph 666 (or by alternative number 616) is all about the perfect man - Jesus.
CCC 666: Jesus Christ, the head of the Church, precedes us into the Father's glorious kingdom so that we, the members of his Body, may live in the hope of one day being with him for ever.
CCC 616: It is love "to the end" that confers on Christ's sacrifice its value as redemption and reparation, as atonement and satisfaction. He knew and loved us all when he offered his life. Now "the love of Christ controls us, because we are convinced that one has died for all; therefore all have died." No man, not even the holiest, was ever able to take on himself the sins of all men and offer himself as a sacrifice for all. The existence in Christ of the divine person of the Son, who at once surpasses and embraces all human persons, and constitutes himself as the Head of all mankind, makes possible his redemptive sacrifice for all.
Personally I also find it ironic that John 6:66, a coincidental chapter and verse number that scholars later add to assist in studying the bibles, is the verse number that tells of those false disciples of Christ who turned their back on Christ to leave him over His teachings on the need to eat His Body and Blood (the Eucharist).
John 6:66 As a result of this [teaching] many of His disciples withdrew and were not walking with Him anymore.
Irenaeus suggested that the number indicates that the beast is the sum of all apostasy committed over the course of six thousand years.
The general consensus with Catholic Eschatology experts is that the Revelation account of the mark of the beast were referring to actual historical persecution of The Church and were not intended to be prophecy but to give courage to Christians to not lose heart. But as has been seen many times in the bible it is often the case that the prophets and inspired writers themselves did not know the full extent and meanings that were conveyed by their words and the meanings did not fully come out for many centuries.