Which one is correct, or if both, what is difference between "has been taken residency" and "has been taking residency"?
EDIT: few more details about the question
I write this phrase in "proof of address" letter for tenants living in my house. I am based in United Kingdom.
Full sentence:
This is to confirm that John Doe has been taking residency at the
property since XXX.
Best Answer
Answering your edited question:
Either the present perfect or present perfect continuous are fine, but in a more idiomatic usage:
and then
or
With since both tenses indicate that the stated action started in the past and continues to the moment of writing.
The difference between the two tenses with since is subtle and not worth worrying about, as they do no affect your purpose.
To wit: the continuous simply stretches out the act of living (or residing) rather than presenting it as a punctiliar event. However, the continuous also is used very often to present a situation that is of limited duration and may soon change.
But again, in your context and purpose, either tense is fine, although I prefer the present perfect (non continuous).