Believe it or not, there is indeed discussion whether this innocuous-seeming staple of Sunday morning post-prayer brunches is ok for consumption under the kosher dietary laws.
Now, there is nothing inherently un-kosher in either lox or cream cheese (as long as no non-kosher ingredients were used in their manufacture). The issue at hand is if they can be eaten together.
Fish is pareve. This means that as far as the kosher laws of meat and milk are concerned, it is a neutral zone. It is neither a meat dish nor a milk dish and can be eaten with either. However, the Talmud warns us not to eat fish with meat, asserting that the combination is unhealthy. This is mentioned in the Code of Jewish Law with the admonition that health concerns are to be treated with even greater gravity than ritual laws. So the accepted practice is to change dishes and rinse one's mouth between fish and meat courses.
So far, our precious lox and cream cheese is safe. But Rabbi Yosef Karo (1488-1575) mentions a health restriction concerning eating fish and milk as well. The subsequent commentaries, including Rabbi Moshe Isserles (1520-1572), argue that this statement of Rabbi Yosef Karo must be an error, because there is neither Talmudic basis nor any other rabbinical precedent for prohibiting milk and fish.
Nevertheless, since Rabbi Yosef Karo wrote that milk and fish should not be mixed, there are those who do not mix them. The Chabad custom is that we do not eat fish together with milk, but we do eat fish with milk products. Even adding a touch of butter or cream to the milk is sufficient to permit mixing it with fish. Certainly then, lox and cream cheese can come together onto any Chabad table.
Rabbi Menachem Posner