Is doing two PhDs a good path?
I suspect that most people who ask about the possibility of doing multiple PhDs are missing something fundamental in what a PhD is and what it's for. This is an understandable misperception because the general populace knows little about what PhD programs are all about, to the extent that even most people who enroll in a PhD program think they know what they're getting into and then find out that what they had in mind is an approximation to the truth (sometimes a good approximation and sometimes not).
In the various fictive media (novels, comic books, television, film...) having multiple PhDs -- often in confluence with a very youthful persona or explicitly pointed out that they were attained at an early age -- is a standard trope for a certain type of characterization: depending upon the genre it can signal super-genius types, intellectuals, or nerdy/socially awkward types. For instance, Professor Charles Xavier graduated from Harvard at the age of 16, and he holds PhDs in Genetics, Biophysics, Psychology, and Anthropology as well as an MD in Psychiatry. This is obviously meant to be a real-world grounding for his vast mental powers. On the TV show House, the youthful medical student Martha Masters has previously attained PhDs in applied mathematics and art history, and the characterization is less positive: they are not trying to suggest that she's a genius per se -- House is the genius! -- but rather that she has an extremely overachieving personality type.
Of course, Charles Xavier and Martha Masters are fictional characters. Moreover I imagine they were written by people who are not so familiar with PhD programs. In real life, having PhDs in Genetics, Biophysics, Psychology and Anthropology is not the hallmark of a super-genius: rather it would mean that the person is pathologically obsessed with graduate study. PhD's are not merit badges -- more of them is not better. Rather a PhD is the necessary and grueling training for a specific type of professional career.
If you are interested in a subject then you read and learn about it. You get a PhD in a subject because you want that subject to be an essential part of your career (which of course ought to imply that you are very interested in the subject). Moreover, although the PhD provides specific technical training in a certain subject, it does not provide exhaustive training: in most cases people spend the rest of their careers continuing to train in these areas. Rather the point of a PhD is to give you enough specific training so that you can henceforth direct your own training and learning. Otherwise put, a lot of what you do in a PhD is learning how to learn. But learning how to learn is actually a rather robust and subject-independent skill. If you learn how to learn genetics, then if later on your interests turn to psychology or biophysics you will be in a much different and better position to train yourself in these areas. In some ways, doing a PhD in one of these subjects and then turning around and doing another is like becoming an internet millionaire who wakes up one day and decides that she wants to sell electronics...so she shuts down her website, gives all her money away, and starts the business out of her parents' basement. (Or maybe it is like what happens in some of my dreams: I am doing high school all over again, and somehow it is not going as well as it did the first time around.) Why are you starting over from scratch?!?
It is true that "academic transfer" makes better sense between some fields than others. If Martha Masters got a PhD in art history and then decided that she wanted to do applied mathematics instead then she would not have been able to "segue" from one field to the other: she would indeed have had to go back to school. But that makes her story a bit sad: getting a PhD is very difficult and very time consuming, but it is not really a feat of strength, and getting two PhDs in two unrelated areas does not show how much of a super genius you are; it shows that you really changed your mind considerably about what you wanted to do with your life and maybe wasted a lot of your time. No matter how smart you are, I don't see how you can get a PhD in art history and then one in applied mathematics without spending at least eight years. You can't just skip to the end and pass the thesis defense by virtue of your preternatural brilliance: there is coursework, residence requirements, and various other mandatory things which necessarily take time. Her character is comic-booky: it is not really plausible that she could have these two degrees and have gone on to be a medical student while still being in her 20's, no matter how brilliant. [Well, "no matter how brilliant" is a little too strong: if she had graduated college at age 16, then the math works out okay...] Moreover, for someone who is that brilliant it would really be sad that she can't seem to figure out anything else to do besides infinite schooling.
Electrical engineering and mathematics are rather closely allied fields. For sure do a PhD in one or the other. While doing a PhD in either one you can choose the amount of involvement you have in the other field, i.e., you can do a very mathematical electrical engineering PhD or a math PhD on a mathematical topic with important applications in electrical networks. Then after you get your PhD you can continue to learn and train in one or both fields as you see fit. It is entirely plausible that you could land an academic job in one department while having gotten your PhD in the other department. This is an ambitious goal, but any academic job is an ambitious goal. Getting the second PhD is unlikely to be directly helpful once you have it, let alone worth the sacrifice of 4-6 years of your life!
Added: By the way, when you mention "mathematics" and "electrical engineering": I think of Raoul Bott, a brilliant and beloved professor whose long career at Harvard ended while I was studying there. I think that his story will be inspirational for many people with this confluence of interests. Check out his bio: he balanced his early interest in these two fields nicely...and only needed one PhD.
Loving two fields is not a good reason to do two PhDs.
The work you do in a PhD is so specific and focused that I guarantee it will "ignore" most of mathematics and most of electrical engineering. Matt Might has a great illustration of this: PhD school in pictures
By the time you finish your current degree, you'll have a better idea of what specific area (i.e., subfield) of research you are most interested in (if you still want to do a PhD at that point). Then, decide whether it would be more at home in a mathematics or EE department, and apply to grad school accordingly.
Are you also interested in getting two jobs or having two families simultaneously? At some point in life people make decisions, long-term decisions that chart out a path for their lives.
The fact that you love both subjects is nothing but positive, and the fact that you have the drive and will-power to pursue higher education in both is admirable. It really is... But here I have lay out some points to consider:
Is it possible to do a PhD in two disparate fields simultaneously: probably.. I reckon it depends on the faculty/university and what they are willing to accept. Normally you can't get two PhDs from the same faculty (at least here it is not possible) and again, normally, universities do not accept other professional commitments while doing doctoral studies (with the exception of medical reasons). That is especially the case if you are officially employed as a grad student and not on stipends.
Is it advisable to go for two PhDs in disparate fields simultaneously: no, probably not. Doing a PhD alone is challenging, and I don't mean "OMG, it's too much work" but rather "OMG, wtf am I doing here!" or "OMG what am I gonna do with my life?!". Don't believe me? Just look at the most voted questions here, a healthy portion of them are about managing the mental burden of PhD studies.
Have you considered doing a cross-disciplinary PhD where you get to play with both subjects, meld them in a pot and make something awesome out of it? ;)
Hope it works out for you, best of luck!
PS - also worth reading: When does one go for a double doctorate?