I do not deny that there are some similarities.
Christianity emerged and grew at a time when mystery religions were popular throughout the Roman Empire. Mystery religions were seen as more vibrant than the more stale state religions. Joining a mystery religion was a personal decision and was not connected to what family or culture one belonged to. Mystery religions sometimes included a sense of personal salvation/benefit/fulfillment.
Despite these similarities, we need to look seriously at the differences.
The thing that tied the various mystery religions together is that there were secret beliefs or rituals that were only revealed to those initiated into the religion. To this day, many of the details of these mystery religions are still a secret. On the rare occasions when details were revealed, there was an angry backlash.
How does Christianity compare to this?
There were no secret Christian beliefs or rituals. Christian proclamation included telling people exactly what Christianity was about in a public context. If people were interested, there was no long process of initiation. The New Testament presents a model of proclamation, belief and immediate baptism.
What about the Lord’s Supper? This was something for Christians but it was not a secret. Early Christians wrote about it (Matthew, Mark, Luke, 1 Corinthians), something that mystery religions did not do.
But didn’t Paul use the word mystery in his writings? Doesn’t that make Christianity a mystery religion?
He made known to us the mystery of his will according to his good pleasure, which he purposed in Christ,to be put into effect when the times reach their fulfillment—to bring unity to all things in heaven and on earth under Christ. (Ephesians 1:9-10)
This mystery is that through the gospel the Gentiles are heirs together with Israel, members together of one body, and sharers together in the promise in Christ Jesus. (Ephesians 3:6)
Paul uses mystery in a very specific way. The mystery that Paul speaks of is the question of what God would do with the Gentiles. Paul announces that the mystery has been revealed in that God was saving both Jews and Gentiles in Jesus Christ. Mystery was important in that it was now revealed.
So mystery religions, as important as they are for understanding the culture in which Christianity began, is not a category in which Christianity should be included.