What to expect when pursuing a second doctorate in an unrelated field

In my experience, nonstandard paths like what you propose are initially viewed with surprise, and with varying degrees of suspicion. That includes admission committees, potential supervisors, fellow students, etc. However, that comes from uncertainty and unfamiliarity and can be shaped by the narrative you supply, and how you follow through.

Basically, people will probably have 3 archetypes in mind when you first contact them. The most positive one is that you are some sort of polymath genius. The most negative one is that you are a serial fail-to-deploy-in-life perpetual student, a model everyone in academia has encountered. And the 3rd is that you want to do research in some unexpected interesting intersection of your old and new field (which you've said is not the case). You therefore need to replace that narrative with a truer, credible, if possibly slightly airbrushed one, something like: you've always been interested in both X and Y, initially pursued interesting topics in X leading to a Ph.D. and start of a promising career, but are increasingly passionate about Y and want to refocus on that. I would be open that you don't expect to directly harness your expertise in X, but that you do expect that your background will help you be a very effective student in Y.

And then you need to back that up with your actions! Your Ph.D. is a credential indicating you have demonstrated the ability to advance the state of the art of knowledge in some field, and that you have learned sufficient academic "tradecraft" to be an independent researcher, at least from a competence point of view. (Footnote: The at-best only partial independence of a post-doc, and indeed the whole post-doc position, is more a reflection of funding realities than of a junior academic's presumed research abilities). So if someone with a Ph.D. and post-docs would approach me, I would expect them to be able to motor through the usual initial stages of a Ph.D. much more quickly than a "new" graduate student. I would expect that they will have done a fair amount of foundational reading in the field already -- that's something that hardly needs an advisor, though an advisor will be helpful in helping identify important strands that may be nonobvious from the outside. And I would expect them to arrive with some well-developed thoughts on potential research topic, though of course we might well adjust it together.

If you were to come with this type of pre-preparation, I'd consider you a lower-risk admit -- as a committee member or prospective advisor -- than someone whose basic skilset is unproven. Conversely, if you were to arrive, with the background you describe, with the typical keen-but-blank-slate mindset a fresh-out-of-undergrad student usually brings, then I would be more worried you belong to the serial-failure-to-deploy archetype instead.

Second, I would suggest you be very open about discussing your financial expectations. Do you need/expect funding (and is it usual in your target field)? Or can you fund yourself by continuing to work in your current field part-time? Or do you have enough saved up from a well-paid science/tech job (and academics in the humanities/social sciences tend to assume all science/tech jobs are well paid....) that you will be self-supporting? All of those are acceptable, just avoid unpleasant surprises from failing to bring up the topic.

Good luck!

It's very difficult to give a general answer, since so many things depend on the specifics of the field, the location, and various other factors.

• Chances of being accepted: that completely depends on where you apply and whether your background matches the requirements for a particular PhD programme and/or a particular supervisor/topic.
• Already having a PhD is definitely not standard. I would imagine that some supervisors will see it as an asset: in some respects, they would get a postdoc for the price of a PhD student. I would imagine that supervisors who are looking for a PhD student who does exactly what they want would prefer a "virgin" PhD student.
• You should primarily ask yourself whether you can afford and are ready to spend a good few more years living on a PhD stipend. You should also be ready to give up on any little academic freedom you have as a postdoc.

A couple more remarks:

• For the record, the field of Computational Linguistics (CL; and its twin field Natural Language Processing, NLP) existed a long time before deep learning became a thing. It's true that it is some kind of hybrid field, but in my experience the vast majority of the research done nowadays is on the technical side (more computer science than linguistics).
• This is why the context described in the link "double doctorate" in your question strikes me as very different: the author did a second PhD which consisted in "writing a monograph on a long-forgotten Scottish intellectual". In my experience PhDs in CL/NLP are rarely this kind of book writing exercise that can be done as a hobby.