“在同一天”，at the same day 和in the same day，都是不规范的表达。

on the same day才是正确的表达，在正规词典上可以随便就能查到。在实际语言表达中，介词on可以省略。例如：

1. People who make mistakes can be given the sack the same day.

人一犯错误可能当天就会被开除。

来自柯林斯例句

2. 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。

来自《简明英汉词典》

3. I want to go there and back on the same day.

我想一天就来回.

来自《简明英汉词典》

4. I attended that screening, later the same day.

在当天晚些时候，我参加了放映仪式。

来自NEWYORKER

5. John just told me that your birthday is on the same day as mine.

约翰刚告诉我你和我的生日是同一天。

来自柯林斯例句

6. Funnily enough, I was born on exactly the same day as my wife.

有趣的是, 我和妻子的生日正好是同一天。

来自《简明英汉词典》

7. Everything he dictated was signed and sent out the same day.

他口授的所有内容都签上名并在同一天寄出了。

来自辞典例句

8. On the same day there was another straw in the ill wind.

同一天,这阵恶风又带来一种不吉之兆。

来自辞典例句

9. We have another event to attend on the same day.

那天我们有其他活动要参加。

来自辞典例句

10. The same day, Philomela read the message woven into the robe.

同一天, 菲罗墨拉读了织在袍子上的口信。