Difference father and mother jew who would like to circumcise a son

Question

If one is married to a non-jew and one begets a son, why is it that when the mother is jewish (but in some cases don't live a Jewish livestyle) can get her son circumcised, but when only the father is Jewish (and is keeping a Jewish livestyle in accordance with the Torah) and want to get his son circumcised it's much more difficult to realise this because the Mother isn't jewish.

Doesn't HaShem teach one should get his son circumcised?

(I used circumcision but maybe this implies to more of the commandments).

Answer 1

Yebamot 23a:

א"ר יוחנן משום ר' שמעון בן יוחי אמר קרא (דברים ז) כי יסיר את בנך מאחרי בנך מישראלית קרוי בנך ואין בנך הבא מן <העובדת כוכבים> {הגויה} קרוי בנך אלא בנה אמר רבינא ש"מ בן בתך הבא מן <העובד כוכבים> {הגוי} קרוי בנך

The son of a Jewish man from a non-Jewish woman is not called his son. He does not give him Mila, because this mitsva applies only to his son following Torah definition.

But the son of "your daughter from a non-Jewish man is called your son," so we need to circmcise him.

Answer 2

The primary commandment applies to the person who is being circumcised. The commandment is that any Jewish male must be circumcised at eight days old. The responsibility of the father is to ensure that the commandment is carried out. If the father cannot do so, then the responsibility moves to other Jewish members of the family.

For example Rambam Milah Chapter 1:1

Circumcision is a positive mitzvah [whose lack of fulfillment] is punishable by karet, as [Genesis 17:14] states: "And an uncircumcised male who does not circumcise his foreskin - this soul will be cut off from his people."

A father is commanded to circumcise his son, and a master, his slaves. This applies both to those who are born in his home and to those purchased by him. If the father or the master transgressed and did not circumcise them, he negated the fulfillment of a positive commandment. He is not, however, punished by karet, for karet is incurred only by the uncircumcised person himself. The court is obligated to circumcise that son or slave at the proper time and should not leave an uncircumcised male among the Jewish people or their slaves.

You ask

Doesn't HaShem teach one should get his son circumcised?

The basic answer is not exactly. As the Rambam says, Hashem commands that one should arrange for his son who is required to be circumcised to get a bris milah. If he fails it falls to the rest of the family or the court.

Halachically, the child of a non-Jewish mother is not related to the (Jewish) father and no such responsibility exists.

If the child is not Jewish, then he is not required to carry out any of the 613 commandments that are not also part of the Sheva Mitzvos Bnai Noach (7 commandment categories non-Jews are subject to). As a result, there is no responsibility for anyone to arrange for a (meaningless) "ceremony". It is only a surgical procedure as with any other non-Jew. If, when the child grows up, he decides to convert, then he would be required to undergo circumcision.

An analogy would be that a non-Jew does not have to eat kosher food and would not be considered as having fulfilled the mitzvah even if he does. Nor would it be a violation of anything if he eats non-kosher food.