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.
Trademark law is actually pretty useful. I say this as someone who is generally quite anti-IP. If you really want to not involve trademarks, just don't enforce it, and you'll lose any trademarks you've got afaik (IANAL, this is not legal advice etc).
What exactly is your worry here? Imagine you've started your project with distinct branding of Foo, and someone comes in and does something you really don't like. Maybe it's adding a gaping security hole in the name of adding a "feature". Maybe it's saying "this project is for Nazis only". The traditional response in open source is to say "fine, you do whatever you want, but create your own fork in a way that won't confuse people". IMO that's fair and square. What happens though when your project Foo gets a bunch of complaints because someone created a "foo" package that secretly replaces all of your keyboard input with swastikas? Trademark lets you say "don't name your package foo, because it's confusing to us non-Nazis".
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.
You might look at the licensing terms for the "Linux" trademark, for one well-known example. There are both formal sublicense terms that businesses and projects can sign up for if they use "Linux" as part of their own trademark, and also stated terms for the use of the trademark without a formal sublicense.
Trademarks are not an issue per se. It is IP. In a world without IP Trademarks would be an issue of fraud, not "Intellectual Property" - and then only for the customer.