That doesn't describe a census or anything like it. There is absolutely no evidence there was a census covering the Roman empire let alone the whole world (as actually stated in gLuke).
However, there was a census of Judea ordered by Quirinius when Herod Archelaus was kicked out in 6AD. And that makes sense because, prior to that time, Judea was a client state so Rome would not have directly taxed it. Once it became a province, it would be subject to direct taxation and, hence, would have needed a census to determine the taxable population.
So, by far the most likely scenario is that the author of gLuke was referring to this census but got his facts a bit wrong. He made way bigger whoppers than that one.
However, there was a census of Judea ordered by Quirinius when Herod Archelaus was kicked out in 6AD. And that makes sense because, prior to that time, Judea was a client state so Rome would not have directly taxed it. Once it became a province, it would be subject to direct taxation and, hence, would have needed a census to determine the taxable population.
So, by far the most likely scenario is that the author of gLuke was referring to this census but got his facts a bit wrong. He made way bigger whoppers than that one.