When we translate the phrase word for word, we get
der gleiche Tag
Which is not wrong BUT it’s only used when the day is actually the subject. Like here:
- The 7th and the 17th of Januare are the same day (for example, both are Mondays)
- Der 7. und der 17. Januar sind der gleich Tag.
The more common use for the same day is to actually talk about when – to indicate a point in time.
And for that, saying der gleiche Tag is wrong because what you actually say is
am gleichen Tag
This is basically much “on the same day”, but in English you can often skip the “on”. In German you can’t and also, it gets fused with the article to “am”.
Here’s an example:
- Thomas and I met Maria the same day.
- Thomas und ich haben Maria am gleichen Tag getroffen.
And if you’re wondering what the difference is between am gleichen Tag and am selben Tag – well, technically, the former is the same “category” of day (Monday, Tuesday,… ) while the latter is the EXACT same day, with the same date.
But much to the chagrin of purist, at least in spoken German, people are not really strict with this and often use them interchangeable.
By the way… this is a general difference between selbe and gleiche. But that’s for another day ;).