Is there any website where you can choose "I am from Country XYZ" and then it shows you a list of countries you can e.g. go to without a visa and the allowed duration of stay. I know I could go to the Ministry of Foreign Affairs website of my country, but I want to see the passport status of other countries as well.
There are many. Wikipedia for example https://en.wikipedia.org/wiki/Category:Visa_requirements_by_nationality, and Passport Index https://www.passportindex.org/. As stated in a comment by @Michael Hampton, always confirm the data found via such sources with an official source to check it’s not out of date.
You can also usually find this information via the Immigration / travel advice pages of many countries eg
for US passport holders it’s https://travel.state.gov/content/travel/en/international-travel/International-Travel-Country-Information-Pages.html
for the UK it’s https://www.gov.uk/foreign-travel-advice
These advisories typically include links to the destination country’s Immigration pages, so they are a good way to find an official source of information about the target country.
Finally, there is Timatic, the system used by airlines, which provides comprehensive information and allows you to search using a variety of criteria eg transit point(s), residency https://www.timaticweb2.com/integration/external?ref=d975cfc59f5c0abd06d16e872198110b&clear=true and the IATA Travel Centre https://www.iatatravelcentre.com/home.htm