There is no certain point in British history where the whole name became a fixed, familial identifier.

We know that hereditary familial names were being used in the 11th century but often this was only amongst the ruling classes and referred to a duchy or similar.

Most land-bound people travelled very little at that time and, compared to modern society, had such a small circle of acquaintances that 'first' names followed by a locational or occupational identifier were enough for identification and communication.

By the 1500s it's thought that about 70% of people were using a hereditary, patronymic 'by-name', largely as a result of increasing populations, wider circles of personal and professional acquaintance, and the evolutions in church record-keeping requirements for births, marriages and deaths. And while we're on the subject of death... taxes. Record-keeping for financial purposes required a "full" name.

It wasn't until the 20th century that deeds poll were required to change one's name from that registered at birth, although the requirement to give other "known as" names for certain legal transactions goes back to the 1500s.

In summary: mostly by the 1700s, certainly by the 1900s... but there is no one date or era and, like many things, it was dependent on your social class and the name you used.

Source:

Personal Names Studies of Medieval Europe: Social Identity and Familial Structures; Beech G T, Bourin M; Chareille P; 2002

Follow up question: is there a similar trend in last names for other cultures? If I'm, say, a Pakistani weaver, would I take that as my family name?

I know this is probably a better question for r/linguistics, but I am quite curious what the historical context there is for this!