I have read on other threads here (as well as on other pages online) that the て in expressions such as どういたしまして, 初めまして or あけましておめでとうございます was not the て-form but rather "a final particle used for a returning remark" (as stated in this thread https://japanese.stackexchange.com/a/13055/54803), where the person who answered names the particle 反問的用法の終助詞.
Could you please say a bit more on that particle, its etymology, what it means and how it is actually used?
て in どういたしまして are different from the other two.
大辞泉 definition only gives the meaning as a question marker: 質問や確かめの気持ちを表す。「あなたにもでき―」「いらしたことあっ―」 (FYI this is normally used by females).
反問的用法 should be a variant of this and interprets どういたしまして as 'What have I done (to be thanked)?'. It is not possible to understand 初めまして or あけまして this way.
The other two are more simply te-form, indicating the sentence continues (初めましてよろしくお願いします).
Etymology can only be guessed, but all of the above (終助詞 or not) seem to have derived from renyo-kei (stem of te-form) of つ, auxiliary verb for perfective (roughly modern た): 〘接助〙 (完了の助動詞「つ」の連用形から).
Note all the expressions can be translated with perfect tense: What have I done?, Now we have met for the first time, Now the new year has come.
反問的用法の終助詞cite this Wiktionary entry as the source, and Wiktionary's definition of this て can be found here (軽い疑問・反問を表す).