I’ve literally done that inside the EU, though to the UK (back before Brexit) rather than Germany - I flew to London and stayed about a month in a hotel whilst looking for a contract there (I’m a freelancer) and more permanent accommodation.

Years later I did the same to Germany, though I only stayed 3 months.

The only requirement is that you either have a job or have the money to pay for the costs of living there (so you can still go without a job, as long as you have the money to pay for a place to stay, food and so on). The reason for the requirement that you can pay your way (either from a job or savings) is because people can’t just move to another EU country to do things like living on the street and begging or living of the local Social Security.

Some countries also have a requirement that you register after 3 months there (for example, Germany), though it’s not any kind of applying to stay, it’s simply registering as living there. This is usually because there are associated obligations for residents in that country, not just in terms were do you pay tax, but in some countries (for example, Germany and The Netherlands) there are things like mandatory health insurance.

In practice as an EU citizen, if you have the savings or the kind of job which you can do in 3 month stints or remotely, you absolutely can hop from country to country every 3 months without having to register with anybody (though I’m not sure how taxes would work - I suppose you would pay them in the last country you registered as a Resident).

If you know the language, if it weren’t for taxes being per country and the rights and duties of Residents being different in different countries (such as the Mandatory Health Insurance for Residents in some countries but not others) hence the requirement to register after 3 months in some countries, the whole thing would be as easy as moving within your own country.