Your problem doesn't relate to statistics.
plot(fit) works because
plot() has the method to draw
survfit class. But your arguments aren't specific class and don't have any information about Drug. So,
plot() connects all points by one line. You need to treat time, surv, and Drug information.
Here is my example:
fit <- survfit(Surv(time, status) ~ x, data = aml) # using aml as example data
df <- data.frame(time = fit$time,
surv = fit$surv, # strata has each number of elements.
drug = c(rep("a", fit$strata), rep("b", fit$strata)))
# plot() draws not data but axes and labs.
plot(log(-log(surv)) ~ log(time), df, type="n", xlab="log(time)", ylab="log(-log(survival))")
lines(log(-log(surv)) ~ log(time), subset(df, drug=="a"), type="s")
lines(log(-log(surv)) ~ log(time), subset(df, drug=="b"), type="s")
points(log(-log(surv)) ~ log(time), df, col=c("red", "blue")[drug]) # to assist you