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How do you correctly punctuate a sentence where it is asked if another person is also someone else?

For example: "Is Greg Dr. Jones?" (which, in this context, is the same as asking "is Greg ALSO Dr. Jones?")

How would you punctuate this sentence?

The way you've punctuated it is fine, but note that the traditional treatment for a shortened word is to use a full stop (AmE: period) for abbreviations (where the end of the word has been removed) but not for contractions (where letters in the middle of the word have been omitted). Hence:

Doctor: Dr

Professor: Prof.

Reverend: Rev.

Reverend: Revd.

Right Honourable: Rt Hon.

Therefore, a more standard (traditional) punctuation would be

"Is Greg Dr Jones?"

[NB: these are stylistic choices and differ according to region and professional context. Wikipedia says that "In American English, the period is usually included regardless of whether or not it is a contraction", whereas the University of Oxford Style Guide 2016 says "Don’t use full stops after any abbreviations, contractions or acronyms". All guides emphasis the importance of consistency: choose the style you want/need and stick to it throughout your document!]

There are a couple of elements in your original sentence that probably cause confusion, so an explanation might help.

Firstly, there's the contraction "Dr". If your style guide says to use a period (or if Greg had been Reverend Jones and you're using the abbreviation "Rev. Jones"), the period might have caused some hesitation due to its location in the middle of the sentence. However, English readers are familiar with such punctuation and recognise it as flagging an abbreviation rather than the end of a sentence, so there's no need to add further punctuation to provide an additional signpost.

Secondly, there's the succession of proper nouns: Greg and Dr Jones. It's true that in English we would normally separate these with punctuation or a conjunction – but in this case the succession is caused by the inversion of subject and verb to form a question: "Is Greg..." instead of "Greg is...". The inversion serves as a flag to let us know that the next component after the subject (Greg) will be the complement (Dr Jones) of the verb (is). Since we already have the "flag", there's no need for punctuation to provide an additional guide to decoding the sentence.

While there's no requirement for additional punctuation, a solution to ensure that the reader doesn't trip up on the somewhat unusual juxtaposition of proper nouns and abbreviations would be to use italic font or quotation marks to emphasise "Dr Jones" - hence:

"Is Greg Dr Jones?"

or

"Is Greg 'Dr Jones'?"

[NB: we can use the single quote marks here because in a sense the doctor's name is being quoted: a longer way to say the same sentence would be "Is Greg also called 'Dr Jones'?" Note that the question mark goes outside the single quote marks here because the quote 'Dr Jones' isn't a question. For more detail on nested quotation marks and how to punctuate them, see my answer to this other question.]

I think it's correct to leave it as-is with no punctuation, but I understand why you might not like it. I have worked around this in the past by making something else happen in the sentence. For example, you could have a dramatic pause:
"Is Greg...Dr. Jones?"
This tends to be how people would say such a thing in real life anyway (at least in my opinion), with a slight pause between the names.

"Is Greg 'Dr. Jones'"?

Single apostrophes inside since you already have outside quotes. They are actually functionally inside quotes, but practically look the same as apostrophes. Just single quote marks " instead of double. Note also the question mark outside of the quote. I prefer this style logically, as well.

Resource on quotation marks in Am and Br Eng: https://writingexplained.org/how-to-quote-a-quote