Can Biologists identify all viruses?
Do physicians/biologists not know all the different types of viruses out there? 


No, Biologists don't know all the viruses that exist out there. There's a lot! We do know many of the ones infecting humans, though, especially the ones leading to the most common diseases. 


  The fact that the doctor didn't know what type of virus she has bothers me. 


Without knowing any symptoms, your girlfriend likely had was some kind of influenza (flu), rhinovirus (cold), norovirus (diarrhea, etc) or respiratory syncytial virus (strong). That's just the most common virus infections. Identifying those is possible (see lab tests for norovirus or influenza), but it's unnecessary in most cases. 

The thing is, no matter which of these she has, the medical advice will stay the same - rest and fluids. So identifying them is just going to add time and money. 

There are some cases where viral identification is done even for these diseases. For example, before giving an antiviral like Tamiflu, which is for example given to pregnant women with the flu. Before giving an antiviral, it should be established that she is indeed infected with the influenza virus. These tests are also done in patients who are hospitalized. In some countries and years, all influenza cases are screened to later know what strains of the virus infected how many people and how it spread. 

As an example, if the result of the test doesn't change the treatment, the influenza test is recommended against by the CDC. With a gastrointestinal issue where norovirus is suspected, nobody is going to make an identification for a straightforward case.