Could your supervisor decide what you post on social media? I am a first year PhD student in Europe, I started in November. My supervisor has recently take a liking into posting on social media and communication-related issues (style, words, hours etc), even for subjects that are not strictly related to our laboratory academic activities. Since he was also my supervisor for my master thesis, I know for a fact that he discovered Altmetrics only a year ago or so.
Problem: today he basically demanded that whenever he posts on social media something that's related to the lab activities, it is mandatory that all members have to like and share on their personal account the post or whatever that will be.
There are other issues with him, but in this case I think that this is a violation of personal space: I should decide what to post or like with my personal social media account, not my supervisor or future employers.
Question: could he even think of making this kind of demands?
Edit: I know that for someone could sound like a rhetorical question (to me, it sure does), but I wanted to hear others' opinion on the subject.
Update: I managed to apply and then win a PhD position abroad in a better university, with a more stimulating working environment and a more professional group.
It was fair on my end to tell my supervisor that I was about to sign a contract and quit the current PhD program: it was fun that in this situation, while I was explaining that I was planning on leaving, my behavior was labeled by him as &quot;unethical&quot;. Furthermore, if we were to collaborate in the future for any kind of reason/project, I would be stigmatized, as in labeled as &quot;unreliable&quot; for the decision of quitting/leaving (a low-key professional threat basically).
First, it would help to know which country in Europe because there are big gaps in the way PhD programs work from country to country. There are untold agreements between supervisors and students depending on the country/university. How far into the program are you?

Second, what social media you are referring to. Is it LinkedIn or Facebook? It would help to know more background, because you can use their features to “trick” a post exposure.  

Third, is he asking you politely, with arguments or he’s just pushing you to do that by imposing authority? If this is the case, there are legal aspects and PhD school directors or coordinators can help with this. Here is why it is important to know which country. In Scandinavia, for example, the system is well organized and you can go and talk to the PhD school director and ask for advice without any consequences. Moreover, the supervision there is not so abusive and if it is (like in your description) then you should consider going to another supervisor (this depends on funding, of course). 

Is he imposing any consequences if you don’t do so? Did you try to talk to him about this? It always help to talk to them and see if you are on the same page. 

Lastly, surely it is unethical and wrong to request something like this, but you should also try to understand why and if this makes sense. At the beginning of my PhD I thought that it would be unethical to add a supervisor/colleague in an article as a coauthor, but later I found out that it made sense after weighting the indirect contribution of that specific person. 

These are the kind of things that you have to solve with a meeting and discuss with your supervisor about what’s bothering you. It’ll also help you to decide if it is worth continuing with this type of supervision. Moreover, if that’s the only thing he’s requiring and he does it politely, you might want to make a “sacrifice”. In the end, you are not sharing fake information or information about something that is not indirectly related to you. You can just share and then delete. Or share only with your group. There are always methods to make a small compromise if you feel like it is worth it from other perspective. If you are at the beginning, then brace yourself for more. You will be surprised how many compromises will need to make to have things running smooth and forward.