In legal language and in administrative law (government) language, you do see same or the same used like that regardless of the variety of English. I only give four country examples below. I think those are enough.
- They did not complete the investigation of (the) same. [same would have to be defined earlier in the text]
It is considered very formal and is used in most varieties of formal, legal or administrative English.
It refers to some matter or subject previously mentioned in a text.
Example from a court case cited online:
"Thereupon, the witnesses were duly sworn and the Court then heard
testimony from the various witnesses for the State and the defendant,
and at the conclusion of same and after due consideration, the
Court does hereby sustain the Motion to have the juvenile jurisdiction
of this matter waived and the proceeding transferred to the criminal
jurisdiction of this Court and to have the defendant tried as an adult
rather than as a juvenile and gave its finding of fact and conclusion
of law concerning same...."
In the paragraph above, the word same refers to the testimony of all the witnesses.
Here's one from South Africa:
It is used six times in the text.
4) That the first respondent be ordered to pay the costs of this
application; alternatively, the first and second respondent, jointly
and severally, be ordered to pay the costs of this application in the
event the second respondent opposes same.
And here is one from Northern Ireland.
[...] relating to disputes over the title to Folio 2974 and the
litigation over many years now in respect of same. Northern
Ireland - same
I recommend not using it in normal circumstances.
Also, for me, this: When you have gone through my notes, give the same to me.
is not really great.
When you have gone through my notes, give them back to me.
And this: The police started the investigation but couldn't complete the same. could conceivably be found in a not-so-well-written police report.
The formal nature of same does not go well with couldn't. But police officers are not known for being good writers.