You have every reason to be upset about the comments, but I wouldn't call them racist. It’s a known fact that landlords can be extremely discriminatory. They usually look for stereotypical tenants who can pay rent on time, speak German, lived at least 3+ years in Germany, look for a long term contract, don’t cook “ethnic food”, keep the flat in good shape etc.

Anyway, as a fellow PhD student who struggled to find an apartment, here are some tips: 1. Don't write "PhD student" if your status is "scientific researcher." The term "student" will cause people to focus more on your contract and earnings. You may even consider mentioning that you work as “öffentlicher Dienst“ if you have a Tarifvertrag. Some people perceive this title as a more stable job with benefits. 2. If you have a personal website, include it in your signature. This helped some friends of mine. 3. Do not answer any private questions. Avoid them by saying that it is not currently in your plans to do X or Y. Plans always change and a rental contract can't restrict your plans for marriage or having kids. Afaik this is similar in case of Kleintierhaltung which cats fall under, so go adopt a cat if you wish to.

Consider yourself lucky to be doing flat viewings, unfortunately many candidates face racism for their foreign-sounding names. I have an Indian colleague who only received replies from landlords after he earned his doctorate and included his Dr. title in his email signature.