What does 管 mean as in "他管我借钱."?
"他(問/求/向)我借钱，但我没借给他" would make more sense, 


'管' has many meanings and it is a shorthand for many terms including  管理(to manage) ,  监管 (to supervise),  管制 (to restrict),  掌管 (to control) and 包管(guarantee) etc..

In '他管我借钱，但我没借给他' the most logical meaning for '管' is 'to manage' which implies 'to count on' or 'to rely on' 


  管 = to manage --> put responsibility on --> counting on --> depend/ rely on


So the translation of '他管我借钱，但我没借给他' should be "He was counting on me to lend him money, but I didn't lend him any"

other example of 管 as different shorthand:

在這處住我管你一日三餐 = 在這處住我(包)管你一日三餐 (you live here and I'll guarantee you three meals a day)

香煙的銷售管得很嚴 = 香煙的銷售(监)管得很嚴 (The sales of cigarettes is supervised very strictly)

我管你去死 = 我(哪)管(得)你去死 (I don't care if you die)