How and when..

[visionary]

...did the Jewish lineage go from the dad to the mom?

Quote from: Tracing Jewish Roots: Jewish Exile Response on Ask the Rabbi

...

 

[Monna]

According to the reference the Rabbi gave, it has been so since the book of Deuteronomy was written. Although Deuteronomy was not written by Moses, the content and intent of these verses (Deut. 7:3-4) may be from Moses time.

Do not intermarry with them. Do not give your daughters to their sons or take their daughters for your sons, for they will turn your children away from following me to serve other gods, and the Lord’s anger will burn against you and will quickly destroy you.​

The verses imply that mothers can have a very undermining and subversive role in the religious upbringing of their children. Having said that it is somewhat ironic that the Israelites' daughters apparently did not have the same degree of influence on their children if they were married to non-Jews. Also, if Jewish women married gentiles, according to the Rabbi, their children would be considered Jews.

"Jewishness" may be passed on through women. But that doesn't make Judaism any less a patriarchal society.