Using the expression "the same" for a previously mentioned item

This is correct usage. "the same" or "same", used as a noun, is a placeholder for the object of the previous sentence or clause; here, "the details of XYZ project". IME it's not common colloquially, but in business communication it's everywhere.

Grammatically it's unexceptionable, but it is rather formal for normal chat (such as email).

It's grammatically correct and acceptable, but still I'd say "the same" in the context of your example sentence is a little ambiguous, and looks like it may be missing a noun. I'd suggest replacing it with "the aforementioned", or even better, "said project".

This kind of use of "the same" is very frequently used in patent texts. For example: "Method for producing a laminated object and apparatus for producing the same"

https://www.google.com/patents/US5985202

"However, we would like to inform you that we are yet to receive ________. So requesting you to please provide the same to us, just in case it is available to you."

"Same" means identical or is used as a comparison that shows something is equal, such as, "the same as". The dictionary says it's an adjective, pronoun, or adverb. In this case, as in the above, it's being substituted for a situation or object, making it a noun. The above 2 sentences was one of the many grammatically incorrect emails I see from Indian colleagues. Try to think about what "the same" is a placeholder for. If it's "the same project", then the sentence would read, "So requesting you to please provide the same project to us, just in case it is available to you." What does "the same project" mean? The same as what? The same as itself? The sentence is unclear at best. Also, if it's a word usage that you only see from non-native speakers, why would you try to find justification for it instead of just doing what is more commonly done? Just replace "the same" with the subject or object you're referring to. It will confuse native speakers less.

It is OED (sense 4a) of the pronounal form of same.

4.a. the same, †that (or this) same: the aforesaid person or thing. Often merely the equivalent of a personal pronoun; he, she, it, they. Now rare in literary use; still common in legal documents; also (with reference to things) in commercial language (where the is sometimes omitted). Cf. German der-, die-, dasselbe.

Examples exist from the 14th century, but the two most recent ones are as follows:

1966 G. W. Turner Eng. Lang. in Austral. & N.Z. vi. 135 A different influence of written language is seen in the use of same as a pronoun equivalent to it, as in ‘put the tailboard up and secure same with a length of wire’ from New Zealand (Wally Crump, 1964), a facetious borrowing of lawyer's English which is quite common.

1973 N.Y. Law Jrnl. 24 July 4/4 The following sentence in a brief is typical of its misuse as a noun: ‘Waldbaum purchased the soda..then stacked it on the shelves in order to sell the same.’