…If someone pronounces the president’s name: DOUGH-nold TRAAHMP.
The first name is just how it looks. The first syllable is one and the same with the word.
I can only describe the phonetic vowel sound in the last name this way. It’s like a blending of the ah in Zahn and the short a sound in Ham. My tongue always reaches the roof of my mouth if I repeat and hold it for at least a couple seconds.
The person saying it like that has Irish origins and is doing it as an affectation, so there is an outside chance the answer is there is no answer.
I don’t think it’s any kind of Irish accent or even an imitation of one. Characteristic of Irish speech is that the “u” in “Trump” would sound like the “u” in “put” (i.e. it is a more closed vowel), whereas the speaker in the OP moves towards a more open vowel, as well as lengthening it.