No, Tacitus was not a Christian. But when a historian speaks of "independent sources", she or he does not mean that people who report something came from distinct groups, but that their accounts are based on unreleated first evidence. This would be for example the case, if we were to find the archives of Pontius Pilate (and make sure that they are no forgeries) which contain a trail record of one "Jesus of Nazareth".
Tacitus not being a Christian makes him not automatically an "independent source" in the eyes of historians. We need to ask, what was most likely his source (and the source of his source, etc.). This is why historians often prefer to use in such circumstances the more precise term "independent tranditions" instead of "independent sources". Two "independent sources" are either an independent primary or secondary (or tertiary, or ...) source unconnected to each other. Discussing what speeks for or against the independence of sources about the same topic is bread-and-butter work of so-called "source criticism".
That we do not know the source of Tacitus' knowledge and probably never will is not unusual, but the rule for the ancient world. We only have a few important primary sources from that era.[1]
What a historian of Antiquity learns during her or his training and what makes a large part of their academic work is to evaluate such secondary (or tertiary or more remote) sources regarding their most likely history of tradition. In this respect Tacitus' account is a very bad candidate of being really independent, since it is extremely unlikely that it was not even indirectly influenced by someone who knows something about "Christ" without ever having come into contact with the Christian tradition. If instead, he would have claimed that his knowledge was based on old documents from the archives of Pontius Pilate, the picture would be entirely different. Though, we still could have our doubts that such archives were really authentic and not Christian "forgeries".[2]
[1] We have almost no autographs of important historic persons from Antiquity. A noteworthy exception are letters from Bar-Kokhba, the leader of the Jewish revolt from 131-136. See: https://en.wikipedia.org/wiki/Cave_of_Letters#Bar-Kokhba_let...
[2] Christians produced a lot of fictional document, such as letters of Jesus and the Apostels, some of which made it into the New Testament. See: https://en.wikipedia.org/wiki/Pseudepigrapha#Biblical_studie...