We don’t actually know much about how the ichthys (“Jesus fish”) became a symbol within early Christianity, but the most likely story isn’t about persecution-- there’s actually more archaeological evidence that the fish started out as an ancient meme. Fragments of early Christian texts from the 2nd century CE show the words “Ἰησοῦς Χριστὸς Θεοῦ Υἱὸς Ϲωτήρ” (“Jesus Christ, Son of God, Savior”) being abbreviated using the acronym ΙΧΘΥϹ, which is also the Greek word for “fish”. Scholars speculate that repeated use of fish and fishing metaphors in the New Testament, Jewish traditions linking fish with water and ritual practices, plus the fish-related miracles attributed to Jesus in the Gospels (and presumably any preceding sources) may have given this acronym added meaning and made it… well… irresistibly memeable.

Public ΙΧΘΥϹ graffiti dating to the mid 2nd century has been found-- there’s even a very early one in the Vatican necropolis, which was then a Roman burial ground near the Circus of Nero. The catchy acronym quickly spread around the Roman world, and by the 2nd century CE, the ΙΧΘΥϹ acronym appears in epitaphs, sometimes alongside images of fish or the “Jesus Fish” symbol as we know it today. The ichthys symbol-- as well as fish-themed art-- can also be found in places where baptisms were performed, and early Christian art depicting baptism often includes fish. By the early 3rd century, Christian writers such as Tertullian were using “ichthys” in metaphor and wordplay even when they were not writing in Greek, indicating that this was a well-known meme even for Christians of different cultures.
Although some historians have speculated that the ΙΧΘΥϹ acronym was used as a way for early Christians to communicate with each other during times of persecution, there is no evidence to support this hypothesis. The Roman government was not a fan of Christianity (they did execute Jesus for treason, after all) but for several reasons the persecution of Christians in the Roman Empire didn’t get widespread traction until the 4th century. Part of this is because Christians within the Roman Empire appear to have had generally neutral or friendly relations with non-Christians. Part of this is because the Roman government was seriously understaffed and spread out over a wide geographic area with a big population and simply didn’t have the resources to mount a campaign against Christianity. One recorded example is from Emperor Trajan’s surviving letters, in which he tells Pliny the Younger (then governor of Bithynia and Pontus, located in what is now northern Turkey) that “Christians are not to be sought out” because it was too much of a nuisance to bring them to trial.
However, there were some notable waves of anti-Christian religious persecution. Under the Emperor Decius, significant numbers of Christians refused to comply with a 250CE edict ordering all non-Jewish Romans to make sacrifices to Jupiter (lots of Christians did actually comply, which caused friction in the early Church). This unfortunately coincided with the outbreak of a mysterious and virulent plague (likely a filovirus or other hemorrhagic fever), which led to Christians being blamed for said plague (they’d refused to do the sacrifices, after all) and criminalized. There is a wide range of estimates for how many Christians actually died as martyrs under this and subsequent spates of persecution.
The next wave occurred at the start of the 4th century CE and backfired spectacularly on the subset of Roman politicians who wanted to get rid of Christianity and spearheaded the effort. This is the “feeding Christians to lions” period which is usually focused on in pop culture. The Romans failed to “get” some core tenets of Christianity and were unprepared for Christians seeing death in the arena as aspirational. Voluntary pursuit of martyrdom was frequent enough that Church authorities issued new limits on who counted as a martyr in order to discourage people from getting arrested on purpose. The martyrs appear to have been great publicity for Christians, as it brought attention to their religion (which had previously been regarded as a superstition) and attracted converts who were inspired by the martyrs’ zeal. So it’s unlikely that Christians needed-- or even wanted-- to create a system of secret symbols to avoid detection by the Roman government. But that doesn’t detract from the fact the “Jesus Fish” is one of the world’s oldest and longest-lived memes.

Reading/Sources
Revisiting the Ichthys: A Suggestion Concerning the Origins of Christological Fish Symbolism [x]
The Origin of the Fish Symbol [x]
Jewish Symbols in the Greco-Roman Period [x]
Martyrdom and Rome [x]
The Roman Empire at Bay AD 180–395 [x]
The Fate of Rome: Climate, Disease, and the End of an Empire [x]