How does one call postgraduate studies in countries where the appropriate degree is not PHD? It can't be PHD scholarship. Can I use "postgraduate studentship" in this sense in official papers? Maybe "postgraduate scholarship" is better or there are some other variants?
How does one call postgraduate studies in countries where the appropriate degree is not PHD?
You need to be precise about which term for the title/degree you would use. In the follow-up comment you reveal that the degree is "candidate of sciences". In Eastern Europe there were such degrees given in the past, in most countries this was one way or another changed to PhD, or similar. In the past, the process towards the "candidate of sciences" (CSc. degree in a country I am familiar with) was called
"I work on my candidature." Just like that. You would also call it
postgradute studies, but see also this question for potential caveats.
However, as Paul Hiemstra points out, you would use this in your system, but when communicating with the world you should use the equivalent term. Most countries have a signed international treaties regarding education and professional degree recognition and equivalence with most other countries (at least in Europe and OECD). In those treaties (usually public material) you should be able to find the exact tables of equivalent degrees, possibly with various stipulations about their use. Look for such a treaty which your country signed with the country where you want to use your title/study term.
Scholarship is usually the funding you get to perform your
This really depends on where you are going to use this term. The goal is to make clear to people what kind of degrees you have. So if you communicate with people from the US, you can just say you are working on a PhD. If you are communicating with people form your own system, just use the title that is appropriate int that context.
For example, until recently the Netherlands did not have the Bachelor-Master system. This did not prevent people from using the MSc title abroad when that was the title people where familiar with.