What does potential PhD adviser want to see from a candidate that is applying for a second PhD/restarting a PhD? I'm about to (not) graduate from a grad school. I've put "(not)" intentionally.
This is my situation. I started working on a project that was new to our research group. The new topic only remotely related to what has been done before in the group and by the supervisor. The project was presented as a hot topic where I can "make my name" and graduate fast. 

As far as I now understand, my supervisor didn't have much experience in this field although he's been around for some time - he is not a new professor. At the time when the project was assigned to me, my supervisor was trying to bring a new faculty (who has extended experience in this field) to the department but it didn't work out - this person didn't get the position, although he did help me with the initial experimental setup and some consultations. I'm really grateful to this person for that.

The topic turned out to be pretty well developed, not much room for wandering around in hope to bang into something big. Besides that I was left completely alone to work on the topic - no relevant classes, no seminars, no advice from my supervisor, no one else in the university working in this field. I ended up making my own experimental setup, fabricating samples, running simulations, writhing data processing scripts and looking for the objective for the project completely on my own. The supervisor's approach was to find something interesting but his planning didn't go any further than that. He never showed in the lab during my PhD.

When I raised my concerns about the direction of this activity he would just either mention some irrelevant anecdotal stories or say that others working in the field have better tools but we have ideas and that's why we are better. 
During our latest conversation he just said "I cannot help you with that." The funding for the project was over and I had to TA classes that are not related to my undergrad field or the current project - getting some knowledge in the subjects takes additional effort. I had to RA initially.

My work resulted in several papers (not with me as the first author) that were from side projects. I also spent one summer doing an internship at a big company. Trying to overcome the circumstances I put more and more work basically working 80 hour weeks (no weekends or holidays). But I find it extremely difficult to reach a goal when this goal has never been even discussed. My productivity went down. When I asked my supervisor about what the objectives of the project are his response was: "to find something interesting." I feel utterly unsatisfied.

I feel completely drained right now, and in a pretty bad mental state. I applied for several post-doc and industry positions without much success. I feel that I have to compete with the applicants that didn't have to reinvent the wheel every time to get something done. One rejection was due to the lack of the first author publications.

During my PhD work I steered the project towards more fundamental research. Right now I'm applying for another PhD in more fundamental field that I studied by myself. It is rather remotely related to the current PhD topic.

Defending where I'm right now is quite doable even with what I have and my current supervisor is willing to write a good recommendation letter for me. My concern is getting a second PhD and funding issues related to this. What would be the best way to approach potential PIs for the second PhD? If a professor reads this post: how does he/she perceive this situation and what he/she would want to know/get cleared before considering an applicant like me?

I'm currently in engineering and looking to continue in physics. Changing the project doesn't appear as an option to me. I raised my concerns about the current project several times and it didn't result in anything. Besides, this is my 5th year and I should graduate (or not). On top of that, the situation as a whole must be changed, I'm not happy staying here.
You should continue to apply for postdoc and industry positions and not consider quitting or applying for a second PhD.  The things you've described here are not atypical for someone working on a PhD.  It is expected you will work independently and in a self-guided fashion.  As a PI, I have little sympathy for a student who runs out of RA funding and now has to TA.  Being a TA is valuable job experience that you should be proud of and having an RA position is a luxury to all graduate students that you should not play off so lightly (it is very difficult to get soft money in grant funds to support RA's in the first place).  Your description of working 80 hours per week on holidays and weekends sounds very typical of a student trying to finish their PhD (I did that for many months leading up to graduation).  Going back for a second PhD would not necessarily increase your chance of success in finding a postdoc or industry position.  It sounds like your are simply discouraged by your situation and have not searched for positions long enough.  Keep at it and stay positive.  Finish your current PhD.