I don't think it's obvious that the Gay Pride rainbow flag actually is the Mark of the Beast. I do think it's obvious that there is a typological correspondence between the two. A Typological Correspondence is when two things are related in the sense that the first thing is the type, and the second thing is the fulfillment. In the Bible there are many types of Christ. A couple of examples:

1. Adam is a type of Christ in that he exercised federal headship over his children (not just his direct biological children). Jesus is the fulfillment, because He is to us what Adam should have been, but couldn't be. For more on this, see Romans 5.
2. David is a type of Christ because he was a godly king. Jesus is also a King, but He is the King who has never failed. Jesus never failed to exercise justice (Ammon), committed sexual sin (Bathsheba), etc. Jesus is what David should have been, but couldn't be. He is the King that Israel always longed for.

In the same way, there are types in the negative sense as well. Rachel Held Evans, for example, isn't The Antichrist, but she is an antichrist. The Antichrist will be like her, in that he will defy God, hate the Bible, and lead people into sin, but he will do so on a much greater scale. Other types of The Antichrist would be men such as Matthew Vines, Arius, Brian McLaren, etc.

With that understood, here's what I mean about the Gay Pride flag:

The Mark of the Beast is intrinsically idolatrous, requiring you to obey The Beast, rather than God's Law. The Gay Pride flag is a celebration of abominations.
You cannot take the Mark of the Beast without worshiping the Beast. You also cannot take the Rainbow Flag without worshiping secular humanism and sexual depravity. The Gay Pride flag is an idol.
You cannot take the Mark of the Beast and be a Christian at the same time. You also cannot be a supporter of gay pride and a Christian at the same time.
Those who refuse to take the Mark of the Beast are removed from society, and not allowed to buy or sell. The Gay Rights movement is determined to remove Christians from society by pushing us to the margins, and to stop Christians from buying or selling by removing us from the economy. This is a regular feature of their activism: exerting economic pressure to cow opponents into submission. Further, literally all of the Christian persecution of the West arises from the purveyors of the Mark—I mean Gay Pride flag.

It's possible (but still unlikely) that the Gay Pride flag actually is the Mark of the Beast, but I think it's clear that the Gay Pride flag is a type of the Mark, and that the Mark will be like the Gay Pride flag in important, significant ways.