According to the links offered at the CitizenGo petition, this claim is being made by Hungarian Prime Minister Viktor Orbán.
“It is a fact that Christianity is the most persecuted religion in the
world today, in view of the fact that 215 million Christians in 108
countries are currently suffering from various forms of persecution,
four out of every five people who are being oppressed because of their
faith are Christians, and in 2015 in Iraq, a Christian was murdered
every five minutes because of their religious beliefs”, the Prime
Google reveals that the figure of "215 million" is being distributed by an group called Open Doors Australia. Google also found me an article where "108 countries" and the ratio "4/5" was used to describe their data. This article appears in the American magazine Christianity Today, which I would describe as a fairly competent source in general, but not useful for data journalism.
Christians were the most harassed religious group, facing arrest, discrimination, and assault in 108 countries, up from 102 countries in 2013 (but falling short of the 110 countries in 2012).
Of the worst 50 nations, 4 out of 5 share the same primary cause. ... The primary culprit in Africa and worldwide: “Islamic extremism,” which was the "main persecution engine" in 40 of the 50 countries on the 2015 watch list, including 18 of the top 20 countries (only 6 of which are in the Middle East).
This could be coincidental if there was a widespread acceptance of the figures of "215 million" and "108 countries", but in fact, Open Doors is the only group making such massive claims.
The Open Doors Australia website does not explain who runs the group or how they arrived at their figure. However, they have a UK branch, which also claims a figure of over "200 million persecuted" and goes into a small amount of detail. It appears they list 50 countries where Christians live in unfriendly societies, and possibly just add up rough estimates of the numbers of Christians in those countries, including India and China.
Open Doors UK is a member of the World Evangelical Alliance and describes itself as "an international ministry serving persecuted Christians and churches worldwide." Another recent answer on Skeptics.SE describes it as a biased source which is a far outlier from other attempts at quantifying violence against Christians.
Furthermore, as you can see from the quotation above, the claim that "4 out of 5 persecuted people identify as Christians" is a severe misreading of the Christianity Today article, and also kind of nonsensical because "persecution" is such a vague word.
I did not find concrete proof that Viktor Orbán is simply misreading the exaggerated claims made in Christianity Today, but given how much of an outlier these statistics are, it seems fairly conclusive that this was a misstatement on his part.