at the same day指在同样的某一天。
例句：
No drama writer of his time can be named in the same day with Shakespeare.
那个时代，没有一位剧作家可以和莎士比亚相提并论。
You cannot have two forenoons in the same day.
一日之中不可能有两个上午。
on the same day 指在具体相同的一天，介词短语，在句子里坐的是状语，即完整的一句话后面或前面加上on the same day。
例句：
A peace treaty was patched together on the same day.
在同一天里,一个和平条约草草地签订了。
I want to go there and back on the same day.
我想一天就来回。
in the same day和on the same day
同一天on
on the same day
例如：You and I were born on the same day
你和我在同一天出生
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