Most church will allow someone to take aslyum from the state (regardless of which country they are in most times) as long is that person is not wanted for grevious crimes like murder or genocide etc... churches usually operate under the sep of church and state and most countries and as long as the church is willing to host these people the state will not move on them...
You raise an interesting question. Maybe it is either some official status of the Church (Lutheran?) in Sweden, which would not be the case in the USA; or maybe it is because Mosques do not offer sanctuary. Islam seems to be a religion of violence and death, not sanctuary and "caritas".
In Sweden, as in most of Europe, the churches are state sposored. They receive tax funds, and are simply a de-facto welfare agency of the government. In such places the so-called "Christian" churches are often even more leftist than the socialist regimes in charge. Muslims, including jihadis, would naturally be harbored by these "churches" since such activity is in keeping with left's ultimate goal of destroying the West. If white Christian Zimbabwean farmers fleeing Mugabe's ethnic cleansing sought refuge there, they would be turned away. Christianity is a thing of the past in most of Europe, and has been so for decades.
In olden days the right of sanctuary was absolute. Even a murderer taking refuge in a church was allowed to leave the kingdom by the shortest route to the nearest port, provided he forfeited all his lands and never returned. This was the common law in England and I suspect many other Christian countries. I suspect the same rule of law may continue in the scandanavian countries. Why not a mosque? Not the same status. The christian nations believed that a person received the protection of God by entering a church. They would not have believed the same of a mosque. Hence, even though the culture may have long gone secular, the ancient right of protection could still exist.
My church has a lot of services for asylum and Convention Against Torture (CAT) refugee seekers. We've a lot of practice helping Chrisitians fleeing persecution from China. I've worked several cases pro bono. It would make sense that those Muslims try to get help from those who know how to help. I can understand why they would want to live in the US but I've seen a lot of false asylum and CAT claims from the Mid East.