Can't get what 'that' means here...

[Flash]

"Enough people were given the same, or similar, names, that we quickly gave up trying to remember all 400."

It's a quote from the book 'My Life in Orange' by Tim Guest. I can't get why he uses 'that' after 'names'. I understand that it means 'so' here but I can't get why. It's the first time I've come across this meaning of that.

[Patria Quiroga]

Well, I understand that "that" here means "which", which you usually use in non-defining clauses.

[Tdol]

I think it's a bit like 'there were so many similar names that we gave up...'.

[Flash]

All right, how often do you use such structures? Do you think I should be able to use them?