Pope Benedict XVI gave apostolic blessings after he “resigned” from the papacy. The apostolic blessing is the papal blessing. Other priests can give it in the name of the pope under certain conditions. However, Benedict XVI gave it in his own name.
The first piece of evidence comes from a leaked letter from November 2017 published by the German newspaper Bild in an article from September 2018. (link to archived article)
The concluding sentence is in German: “Mit meinem apostolischen Segen bin ich Ihr [signature]” Or, in English: “With my apostolic blessing, I am your [signature]”. You can see the pronoun “meinem”, which is “my”. He says it is his apostolic blessing. He does not give it “in the name of Pope Francis”.
On January 13, 2020, the Twitter account of Cardinal Sarah posted letters Benedict XVI wrote to Sarah. At the end of one of the letters, dated September 20, 2019, Benedict XVI. writes in Italian: “Seguendo la Sua domanda imparto la mia benedizione apostolica a Lei, ai Suoi collaboratori e alle persone a Lei care.” (link, link to archived post) In English: “Following your request I impart my apostolic blessing to you, your co-workers and the persons dear to you.” (link to an archived image of the letter)
This shows Benedict still considered himself pope in some sense after he “resigned”.