Why did Muhammad allow Muslim men marry to marry Jewish women?

"they have to claim the mother's religion"

This is incorrect.

Not that they "have to claim it" but that "they are Jewish in fact". i.e. Judaism defines them as Jewish at birth.

Also, I don't find it surprising that one religion would have differing standards than another religion.

Islam is patriarchal. The men dominate and women are to submit to the man's bidding. A Muslim man can marry women of any religion or background, but she will be expected to convert to Islam and follow the leadership of her husband.

Women on the other hand are forbidden to marry anyone other than a Muslim man. Any women who does so would greatly dishonor her family.

Muhammad was the type of guy that overran cities and countries and forced people into Islam whether they wanted it or not.

He just assumed Muslim men would do the same to their Jewish or Christian wives.

Because in Islam the Muslim men will be the leaders in the family SO THEY CAN MARRY JEWISH AND CHRISTiAN WOMEN so they can give influence to their wives and children to become Muslims