While there are similarities between different types of anti-religion bigotries, in one respect antisemitism is unique in that it is bipartisan. Generally speaking, you find Islamophobia on the right and Christianophobia on the left. But antisemitism can be found on the left and the right.
Although antisemitism is bipartisan, it looks different depending on whether it is on the left or right. Antisemitism on the right sometimes reflects a racial animus and at other times there are religious elements in that Jews are seen as “Christ-killers” On the left, it takes on more political dimensions whereby Jews are seen as oppressors and engaging in genocide. There are questions on whether Israel has the right to exist.
So even if antisemitism is bipartisan, you will not see the antisemites on the right work with those on the left. They may have similar goals in the humiliation and/or elimination of the Jews but they have different reasons. Indeed, I suspect that the antisemites on the left and those on the right do not like each other.