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Update: I decided to present my previous experience and I'm trying to get the most of it with my manager. Once I iron out the last few bugs in my current project, I should be able to get more interesting tasks for what my manager is planning for the next features and and a plan with objectives to meet for a promotion.
Now that I cooled off a bit I see that I exaggerated the issue. In the end, I can't choose Adam's behaviour nor my manager's, and confronting either could have only worsened my situation. I will try to get the new title that way, but I am also looking for other opportunities in the meantime.
Though all answers raised valid points, I can only choose one and it was the one that I think is the most complete.
Thank you very much for the insightful answers and the general constructive criticism.

I joined about nine months ago a company where I started a new position in a computer science R&D department, which itself properly began several months before that.

In this department are three other people related to the situation I'm in, let's call them Adam, Bob and our manager.

Adam started working a few months before I was hired, in continuation to his internship. This is his first job, and he holds job title 1, like me.
Bob joined on the same day I did, and has job title 2, which is slightly better paid and is more prestigious. He has a PhD, unlike Adam and me, which is the main reason we were given for the difference between our titles, as our specific tasks are rather similar. He recently resigned and is serving his notice period. He will leave soon, and his position is currently empty.

The tasks performed by Adam and me do not fit our title, as the industry standard would call it job title 2, the one Bob has. Until now, I was trying to get title 2 instead of title 1. I don't particularly care, but it would be better for my career to have a more esteemed title, and it is the correct description for the job I do. Adam is, of course, doing the same, but he appears to have beaten me to it.

Probably because Bob has quit, Adam told him he asked to get his position after he leaves, and will be granted it starting January. Adam did not tell me that directly, nor did my manager. I have only heard it from Bob, but he is a friend that I trust.

After learning of that, I arranged a meeting with my manager to tell him that my title does not fit the tasks I perform and that I wanted to get position 2, but I did not mention that I knew Adam would get it. I was told that as of now, this position required a PhD but that I was a title 2-to become.
  The discussion drifted off, and I managed to talk about a prior experience during my last internship, which mirrors almost exactly the vision my manager has for the products we are working on. I planned a meeting with him where I will talk more about it at the end of the week.

However, not having a PhD does not seem to stop Adam from getting that position, which hurts quite a bit the trust I had in my manager. I will know for sure if Adam does get promoted at the start of the new year, but until then, how should I approach the issue? Should I ask my manager if he confirms it? If there really are such double standards, I think I will just look for a new job somewhere else, as I am not willing to put up with blatant hypocrisy.
I figured sharing my experience would be a good chance to get assigned to higher-level tasks, but now I'm having second thoughts, and I would rather not give anything more than I need in this job if I truly do not have a future in the company.

My plan for now is to confirm with my manager that having a PhD is truly required for title 2, and if he does, to tell him that I heard Adam was getting promoted in spite of not having it. Is it a good idea? Can I handle it better?

This is happening in France, though I doubt it matters.

I joined about nine months ago a company where I started a new position in a computer science R&D department, which itself properly began several months before that.

In this department are three other people related to the situation I'm in, let's call them Adam, Bob and our manager.

Adam started working a few months before I was hired, in continuation to his internship. This is his first job, and he holds job title 1, like me.
Bob joined on the same day I did, and has job title 2, which is slightly better paid and is more prestigious. He has a PhD, unlike Adam and me, which is the main reason we were given for the difference between our titles, as our specific tasks are rather similar. He recently resigned and is serving his notice period. He will leave soon, and his position is currently empty.

The tasks performed by Adam and me do not fit our title, as the industry standard would call it job title 2, the one Bob has. Until now, I was trying to get title 2 instead of title 1. I don't particularly care, but it would be better for my career to have a more esteemed title, and it is the correct description for the job I do. Adam is, of course, doing the same, but he appears to have beaten me to it.

Probably because Bob has quit, Adam told him he asked to get his position after he leaves, and will be granted it starting January. Adam did not tell me that directly, nor did my manager. I have only heard it from Bob, but he is a friend that I trust.

After learning of that, I arranged a meeting with my manager to tell him that my title does not fit the tasks I perform and that I wanted to get position 2, but I did not mention that I knew Adam would get it. I was told that as of now, this position required a PhD but that I was a title 2-to become. I also mentioned that Adam would be interested and that he was deserving too. The discussion drifted off, and I managed to talk about a prior experience during my last internship, which mirrors almost exactly the vision my manager has for the products we are working on. I planned a meeting with him where I will talk more about it at the end of the week.

However, not having a PhD does not seem to stop Adam from getting that position, which hurts quite a bit the trust I had in my manager. I will know for sure if Adam does get promoted at the start of the new year, but until then, how should I approach the issue? Should I ask my manager if he confirms it? If there really are such double standards, I think I will just look for a new job somewhere else, as I am not willing to put up with blatant hypocrisy.
I figured sharing my experience would be a good chance to get assigned to higher-level tasks, but now I'm having second thoughts, and I would rather not give anything more than I need in this job if I truly do not have a future in the company.

My plan for now is to confirm with my manager that having a PhD is truly required for title 2, and if he does, to tell him that I heard Adam was getting promoted in spite of not having it. Is it a good idea? Can I handle it better?

This is happening in France, though I doubt it matters.