I am a PhD student in chemistry and my funding is specific for a certain project. This funding also funds another PhD student, a biochemist, with a different advisor (both our advisors are very well-respected). We're both in the 3rd year of our PhDs.

The project is now in its sixth year. The distribution of work is as follows:

-Some post-docs had done the first 4 years of this work
-The other student provides the animal samples and some biochemical test
-I do the "novel test". This test is labour-intensive and time consuming. I very often work on holidays, while the other student doesn't even answer mail after hours, despite having a technician to help her.

The problem is the authorship of the papers. We are now going to start our third sub-project on this work:

-Paper #1 [last year] was primarily the post-docs work, with authoring PostDoc*, OtherStudent* (where * denotes equal contribution)
-Paper #2 [last year] was mostly my work, with authoring me*, OtherStudent*. I was extremely heart-broken and wanted to quit, but my advisor assured me that this would not happen again
-Paper #3 [this year] is the problem. The other student sent a new batch of samples in for a new paper. My advisor said that it is very likely that the other group will ask a paper co-first authored as: OtherStudent*, Me*, since the other student had not been in the first position of the co-first-authorship yet.

I believe it is unfair that I have fewer papers than this other student despite doing much more work. Indeed, this other student has time to work on other things and get still more papers published, where I am working flat-out on this effort.

My advisor and I argued and I told him that I want to switch to a different project, but he said I can't because my funding is specific to this project. I'm thinking to quit and apply for a new PhD somewhere else but I have spent 2 years and 2 months in my PhD and I believe I will also have problems in the new place (hopefully not as bad).

Am I right to be angry? How should I approach this situation? How do people in academia actually look at co-first authorship? Is the order important?

You may be/feel angry but imho you shouldn't. Just because you are doing 'different' work to the other student does not make you more important/first author over them when it could be their idea/study design.

Authorship is fun issue and I can feel the frustration. You also have the added fun of being in a multi-disciplinary linked PhD project with no clear leader. Your position would always have authorship issues, so not your fault.

Though has the other student got any first name papers? I mean this is their PhD as well. If this work/paper is key to their thesis it would look very bad if they don't have a first name paper. Even if you did all the work they still need to write up the biochemistry side and defend their work, which is very hard if you have both first name papers.

Honestly, I think your supervisor shouldn't have promised you that. He should have known this and your problems. In hindsight there should have been a more equal division of work for a linked PhD, which is not your fault. You probably have made this clear and can potentially leverage this to "politely" ask to get your name on a side project.

Since day one my supervisor has made clear that I shouldn't share my raw data until authorship has been decided. I think that might be applicable for you. If you are clear about your future work that you want first authorship they might be willing to do more of the grunt work. As the person who controls the raw data controls the paper. But this is only applicable in future.

Write this paper yourself. Seriously, if you write the first draft with all the data and your name first. As you have taken the initiative and are leading the work, they have to argue against you which is harder. I would recommend making it clear to the other group that you are unhappy and want a better authorship split with regards to lab work.

BTW, if you have no intention of staying in academia and don't care about relations. You can email the journal and make a complaint about authorship. Most journals will immediately freeze the process until authorship is fully agreed.