Nice you gave us the example. I didn’t realize that NameはNameで meant the same name in both. I was reading it as AはBで instead of AはAで
I think your translation is pretty good. This expression mostly means that what you are going to say next is typical, unique or characteristic of A. Depending on the sentence it could either get an “as expected” nuance or a more “in its own way” one.
A nice way to replace it is 「AはAなりに」, which is a lot easier to search on internet I guess.
I couldn’t find a nice grammar website discussing 「AはAで」directly, but there is this Stack Exchange question and also this paper (full Japanese) on expressions with repetition in Japanese that briefly mentions it, saying you could either see it as a conjugated form of 「AはAだ。」(read the paper) or understand it as having the same meaning as 「AはAなりに」
2.3.1.「A は A で」
「A は A だ」の活用形とも考えられるが、「A は A だ」の用法とは異なる意味を表して
いるため別に分類した。
（34）学生は学生で、社会人は社会人で充実してる内容が違う。
（35）馬鹿は馬鹿で一生懸命生きてるんだから。
いずれも、「A は A なりに」の意味で用いられている。