Is there a copyleft equivalent for trademarks? I'm thinking of starting a project with distinct branding but I want everything to be based in FOSS principles.
This isn’t how trademarks work, unfortunately. A trademark is something that only your business gets to use. Your unique use of the trademark means anything using it comes from you.
You can grant others the ability to use your trademark. Without that approval process that has to go through you, anyone could use it, though. If you don’t attempt to enforce your trademark, you could lose the mark depending on the extent of the infringement. Different folks will tell you different things there.
I wouldn't say unfortunately. Trademark is an important customer protection mechanism, because it allows you to know what to expect from a product or service - even communist countries have trademarks for this reason. For example, if some browser is called Firefox you have the expectation that it allows you to install an ad blocker and if it's named Chrome you have the expectation it... backs up all your personal data in Google's servers, for clothing it's important to know if you are buying something from a brand that makes durable clothing or if it's discardable crap like Primark, and for professional products the difference between a reputable manufacturer and a bad one can literally mean life or death. What's unfortunate is that companies like Disney and Sanrio commodified trademark by building brand cults.
In most cases you don't want one. It can make forks confusing and lend malicious actors more credibility than they deserve.
Copyright controls the code. Trademarks are the recognisable names/icons that identify a project.
Thank you everybody for your feedback, this has been helpful.
Others gave good reasons against, but I'd like to find a way through. Without knowing much about your project.
What if you classify it as art, whatever it is? Others are free to use, change and copy it, as long as they themselves allow the same with theirs.