This is not true. The statement in the question is perhaps a wrong oversimplification of the following facts:
Non-Muslims are not punished for drinking wine, as long as they do it privately. The reason for this is that wine is permissible or even required in their religion and such private religious concessions are granted as part of the treaty of Dhimma.
Some madhabs allow for non-muslims to judge themselves in their private matters in which there is no Muslim affected. And some allow for Muslims to refuse passing judgement in the disputes between non-muslims - in which case they are free to instead get a judgement from their own rabbis and priests. This is based on:
فاحكم بينهم أو أعرض عنهم
Judge between them or turn away from them
— Quran 5:42
However this concession is disputed, since this verse was revealed regarding the Jews of Medinah in the early period, who were not Dhimmis. And there are traditions that it has been abrogated by 5:49 which does not give this choice.
Some minority opinions deviate on specific matters regarding legal punishments on non-muslims. For example the Malikis according to one saying do not impose the punishment of Zina on a non-muslim. While the Hanafis dispute whether non-muslims can be stoned to death for adultery. And similarly there are a few other aberrant opinions. However this is because of their interpretation of some ahadith and traditions and not because of what non-muslim scriptures say (The Torah actually prescribes execution or stoning on the Jews for adultery).
There is an incident where the Prophet passed a sentence on some Jews and asked them for the law of the Torah. However the law of the Torah in this case is the same as the law in Islam, so it is not evidence on anything where the Torah might deviate from the Quran.
If a Muslim ruler judges between non-muslims he must always judge according to the laws in the Quran and he can not judge according to un-Islamic sources, including the scriptures of the non-muslims when they contradict the Quran:
وأن احكم بينهم بما أنزل الله ولا تتبع أهواءهم واحذرهم أن يفتنوك عن بعض ما أنزل الله إليك
And judge, [O Muhammad], between them by what Allah has revealed and do not follow their inclinations and beware of them, lest they tempt you away from some of what Allah has revealed to you.
— Quran 5:49
فاحكم بينهم بما أنزل الله ولا تتبع أهواءهم عما جاءك من الحق
So judge between them by what Allah has revealed and do not follow their inclinations away from what has come to you of the truth.
— Quran 5:48
وفي جميع الأحوال إذا حكم القاضي المسلم بين غير المسلمين لا يحكم إلا بالشريعة الإسلامية
In all cases, if a Muslim judges between non-Muslims he will only judge according to Islamic law
— Mawsoo’ah al-Fiqhiyyah
Hence it is not possible for a Muslim judge to use the non-Muslim's holy book for sentencing.
Further, as far as the Hudud punishments are concerned, scholars have noted there is no distinction if the perpetrator is a Muslim or non-Muslim - both get the same punishment which is obligated in the Quran:
وأما الذمي فهو فيما سوى حد الشرب كالمسلم إجماعا
As for a Dhimmi then other than the legal punishment of drinking he is like a Muslim by consensus
— Jawharah an-Nayyirah
وأما الذمي فيجب عليه جميع الحدود اتفاقا إلا حد الخمر
As for a Dhimmi it is obligatory to impose on him all legal punishments and this is agreed upon, except the Hadd of drinking wine
— Al-Bahr ar-Raiq
The Quranic verses on the punishment of fornication (24:2), theft (5:38), robbery (5:33), slander (24:4) etc. are general and their wording does not differentiate between whether the perpetrator is a Muslim or Non-Muslim. Hence they apply equally to non-muslim citizens despite the fact that many of these punishments are different from the Bible.