Is it possible to submit a PhD thesis with two authors?

It is definitely possible in the sense of "It has been done." However, the only example I know of is this thesis from linguistics: http://dare.uva.nl/search?arno.record.id=123669 (Groenendijk & Stokhof 1984)

Whether it is possible in the sense of "You could do it." will depend on your university's regulations concerning these matters, as well as your advisors and committee's preferences. They are the people you should talk to about this.

No. The PhD thesis consists of your personal contribution to the field and is a demonstration of your expertise.

This presented a problem for a pair of fellow PhD students while I was working on my degree (in the sciences, in the United States). They published numerous impactful papers a co-first authors, but thesis committee members had many objections for the student had the misfortune of presenting second. They had to clearly delineate their personal contributions in the thesis itself, where each had to have enough to individually merit the degree. This was a complete headache for both of them and ended up being a significant amount of extra work.

This merits a discussion between you, your advisor, and your colleague before you get any further in the proposal/thesis writing process.

Edit: As a few have mentioned, it seems like there are some instances where they are accepted - but the overall trend is that they are very uncommon. Do to the many reasons listed in the multiple responses, they are not recommended in general even if allowed by your university. In any case, you both would still need to prove each of your personal contributions are deserving of a PhD.

It happened to me to a lesser extent: a small part of my PhD research was in collaboration with another PhD student in another continent. When we were about to write our thesis (more less at same time) our adviser had to come to a mutual agreement on what part would be in my thesis and what part to my colleague's thesis. Even if it was not against the rules to include the same work, it definitely wasn't advisable to do so to avoid any potential awkward situation in the future.

Even if the adviser agrees, and even if it's technically not against university rules it could potentially raise many red flags in the future for someone that catches the situation (and I think it will be very easy for someone to spot it). I would strongly advise not to.