I've admired @BernieSanders' courage for 20 years and loved getting to know him this last year on the trail. —Pete Buttigieg
It seems to mean "in the past twelve months" according to my research.
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Is “this last year” proper English?
Yes, it means "in the 12 months prior to today"
This contrasts with "last year" which [currently, and in most contexts,] refers to the whole of 2019.
Have a look at the examples given in Google Ngram Viewer.
I came across a very similar phase while reading Pride and Prejudice.
“Mr. Bennet, how can you abuse you own children in such a way? You take delight in vexing me. You have no compassion for my poor nerves.”
“You mistake me, my dear. I have a high respect for your nerves. They are my old friends. I have heard you mention them with consideration these last twenty years at least.”