I have refugee status in Romania, a temporary residence of 3 years and refugee travel document (Convention 1951). I traveled from Romania to the Republic of Moldova using my Romanian travel document. When I needed to make a declaration from a notary (and to receive money from Western Union in a bank), they would not accept my Romanian document and told me that they required a passport or other identification document, that my Convention 1951 document can only be used for travel purposes. Immigration authorities in Romania told me that I have the same rights and obligations as Romanian citizens. When and how can I use my Romanian travel document?
Immigration authorities in Romania told me that I have the same rights and obligations as Romanian citizens
A kind sincere sentiment, but not practically true. Obviously you are expected to obey the laws, and are given many of the same freedoms and access to state services, but there is a significant and critical difference between being a citizen and not being a citizen. One of those differences has to do with access to a passport.
When and how can I use my Romanian travel document?
For exactly the purpose it was meant for: to be able to travel in and out of the country from which you have been granted refugee protection.
The 1951 convention established that as a way for refugees to travel abroad with some limitations. Signatories to that convention (such as Moldova) have agreed to accept it for these purposes. However the convention does not state that any other country must recognize it for any other purpose than for travel, even a purpose for which an actual passport would often be used (proof of indentification for example). That is strictly up to those countries to decide for themselves, and since they may have no specific policy on the matter, it may be left up to individual organizations to decide for themselves. Even Romania, the country that has given you protection and issued the document, technically doesn't have to consider this document as firm proof of identification or require anybody else inside the country to do the same.
This is a limitation you may have to live with for some time.