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The question stem DOES NOT tell us that Y is either a Single Digit No. or a two digit number..

(1) The unit digit of y^2 = 1 ... Y therefore could be any one of the following numbers:

1, 9, 11 (11 x 11 = 121 , unit digit of 121 is one), 19, 21, 29....

So statement One is not suff.

(2) The units digit of Y does not equal 1.

Not suff.

Taking 1 and 2 together we can remove the following numbers from the list :

1,11,21,31 etc ..

This still leaves, 9 , 19, 29, 39..The squares of all result in a NUMBER whose UNIT digit is 1 ..

Why is the official answer ASSUMING that Y has only one digit when it is not mentioned?

Read the question carefully. They are asking the units digit of Y. Not the number Y itself. 9, 19, 29, 39,..... are all values of Y; agreed. But the units digit will always remain 9. Thus, answer is C.
_________________

Agree with TH03 response, we need to concentrate on the unit digit as the questions ask for its value in the question.

(1) The unit digit of y^2 = 1 .

This comes down to either to 1 or 9 as unit digit (the number could be 1,11,21 or 9,19,29 etc) Not sufficient (2) The units digit of Y does not equal 1. This could be anything from 0, 9 except 1 Not suff.

Combining both of them we reach the answer is (C) as 9 is the last digit.

But the question doesn't specify that y has to be an integer. For example y could be 21^1/2, in which case it's unit digit would be 4, or y could be 9, so I think E should be correct.