Mention two-body-problem in PhD applications? If so, where?

I would not do this. For faculty applications there is an argument to be made that alerting the search committee about the two-body problem gives them time to come up with solutions. A faculty candidate is a large enough investment on the part of a department/university that spousal hires/accommodations are a reasonable thing. I have never seen a PhD student, however, who is so good that the admissions committee would give the idea of offering a place to the trailing spouse a second of consideration. They might, however, decide you are not worth the effort if they know a partner is involved.

I had the exact same experience two years ago, also both being in Computer Science in slightly different fields.

In our case our chances were pretty similar as well. However, your sentence ('If you admit me, please also consider admitting my significant other Jane Roe') makes it seem your chances are much higher than your partner's.

If your chances are similar I would mention it in convenient places. For instance when mentioning a project together, talking in person with professors or when a university has a special place to put it (I remember U.Washington had a special field), always remarking that your partner has a similar/better academic level.

If they are not, you can always apply to different tier universities nearby (for example Boston has lots of colleges). Moreover, you should think about how strong the requirement of being in the same university/city is for you. In our case we ended up together in our top choice so there was nothing to reflect upon. However, looking backwards there's two things that I didn't expect that may be useful to think about:

• Having your partner with you makes it much easier to handle day-to-day research's ups-and-downs.
• PhD life has a lot more flexibility than expected: choosing a flexible advisor you could travel and spend multiple days a month together. Specially because in CS you can work from home/abroad.