I can think of two ways of interpreting the second sentence, but it is still different from the first in either interpretation.
Depending on context, the following interpretations are possible:
1. Give me another two shirts.
a) Give me two more of the same type of shirt.
→ If you've already given me two shirts, I will now have four shirts.
2. Give me two other shirts.
a) Take these two shirts back and give me two different ones instead.
→ Once I've decided which I like, I will end up with two shirts.
b) Give me two more shirts, but make them different from those you've already given me.
→ If you've already given me two shirts, I will now have two shirts of at least one type and another two shirts of at least one other type.
But regardless of the context and how the second sentence is interpreted, another and other mean something different:
- two different as replacements
- two different in addition
Note that of the same type is somewhat vague too. It's possible that the type of shirt itself could be random. In which case, there would be no effective difference between 1.a) and 2.b), and it would be theoretically possible for every shirt to be different from the rest. But that would be a practical result rather than one of semantics.