The most unmotivated people I have ever had the displeasure of having to manage were people I had no ability to fire or promote. There is basically nothing you can do yet higher management still holds you responsible for the lack of results.
In addition, don't just give up. Keep talking and helping out. It could be advice on how to stay motivated or at least not demotivated, carving out space for interesting projects or developing new skills, or even helping that person find a new opportunity outside the company.
Dealing with unmotivated unproductive people you have limited ability to actually manage is beyond my experience.
Motivation is the responsibility of the employee. As a manager, if your performance is faltering and it's because you're not motivated, I will tell you clearly my perspective, and ask if that is the case. However it's on you take corrective action. I will do everything I can to help you, but it's your responsibility, and not fair to the other team members who are motivated if
1. I am focusing my energy dealing with issues caused by your lack of motivation.
2. They're working extra to ensure success when a team member is unmotivated.
In smaller orgs with no HR, it's probably easier, but larger companies have a specific process.
Though to your point, I would not consider myself someones manager unless I had the option to hire, fire, or promote them.