No, Cessario is quite explicit about this: “Both the law of the Church and the laws of the Papal States stipulated that a person legitimately baptized receive a Catholic upbringing.” Not merely a Christian upbringing, but specifically a Catholic one. In terms of canon law and the law of Vatican City, what mattered about Mortara’s case was not that the Mortaras were Jewish but that they were not Catholic. Though it’s hard for me to believe that the actuating motive here wasn’t antisemitism, if David Kertzer is right in his book on the case, Pio Nono might have been even stricter with a Protestant family:
Events of 1848-49 only strengthened Pius IX’s opposition to the idea of freedom of religion. He was committed to the principle of the Catholic state, one in which any other religion had to be viewed with suspicion and closely regulated, if not banned. This principle extended not only to the Jews but to other Christian denominations as well. Indeed, the Pope was more favorably inclined toward the Jews, who represented no threat to the Holy Church, than toward the Protestants, who did. To the complaints of those who said that the Jews were poorly treated in the Papal States, the Pope and his defenders could argue that, on the contrary, they were accorded privileged treatment, allowed to have their own synagogues and practice their religion undisturbed. By contrast, Protestants were not permitted such freedoms, and Rome itself had no real Protestant church, other than a converted granary outside town used by diplomatic personnel and other foreigners. Papal police stood guard at its doors to ensure that no native went inside.
There are of course legitimate arguments to be had about whether true Christian faith is compatible with the liberal order, whether separation of church and state is a good idea, what Pio Nono’s true motives were, and so on — but there’s no doubt that the politico-theological principle at stake in the Mortara case does not concern the relations between Christians and Jews but rather the relations between the Catholic Church and everybody else.