I would use a dictionary with a bit of indirection thrown in. When you want to use dynamic variable names that's a hint that you should not be using actual variables but instead some sort of data structure like a list or a dict.
Try using two data structures: a list called variables which stores the names of the variables, and a dict called values which stores their values.
variables = ['y', 'x', 'xE']
values = dict((name, None) for name in variables)
files = [measurable, time, timeErr]
# PROBLEM 1: Assignment problem
#
# Trying to do:
#
# var[1] = files[1] so that if (y == measurable): print("it works")
# var[2] = files[2] so that if (x == time): print("it works")
values[variables[1]] = files[1]
values[variables[2]] = files[2]
if values['y'] == measurable:
print("it works")
# GOAL TO GET ASSIGNMENT LIKE, data is in files "y, x, xE":
for name in variables:
variables[name] = open('./'+name).read()