Are framework names register trademarks?

It depends on the company who own TailwindCSS is trademark sensitive or not.
Some company just don't like anyone using their product name and some just ignore unless you try to mislead customer.

It can be problem for you if you're using it in your product name. Also you can check if they have trademark or not at WIPO DB.

Putting the name in your name is a silly concept. In effect, any work you do will be building their brand not yours.

I recommend coming up with your own name and tailwind tagline (which you can change later)..

e.g. SuperCool (name) -- Tailwind Components (tagline)

Disclaimer: I'm not a lawyer, and I don't play one on the internet. This is not legal advice. In my own personal experience, this is a complex area of law and you should consult with an attorney specialized in intellectual property. This is especially true if you're considering naming your product or service in a way that could in even a remote way be construed as causing a liklihood of confusion with another product.

In the US, if you name your product/service in such a way that is likely to cause confusion in the market, it is called "trademark infringement" or "passing off" and the owner of the mark can not only force you to stop using the mark, they can disgorge you of any profits; they can take all profits and could seek special damages if your use of their mark caused their business to lose business, money, good will in the market, or if it caused any other business losses. These rights don't just apply to registered marks. The United States is a "first to use" jurisdiction and at a high level that means that a trademark does not necessarily have to be registered in order for the owner / inventor of the mark to have rights to that mark; these are called "common-law" trademarks.

To give you an example of how low the bar is to establish a common-law trademark is, I'll use the example of a notable domain / UDRP case. It's worth noting that the plaintiffs application ultimately failed, but this case is somewhat well known because it really demonstrates the very low barrier to establish having rights in a mark (at least in a UDRP case). You can refer to the case of blendid.com which you can find here. In this case the plaintiff simply asserted that they had used the mark on just three kiosks in California, had conflicting evidence of their use (including a formal trademark application with the USPTO that said they had not yet used the mark in commerce) and the panel still found that the plaintiffs "use" met the legal requirements to have rights in the mark.

You are correct that there are fair-use allowances for using a trademark, such as in a reference or in criticism, but using the mark in the name of a product wouldn't likely fall under fair use doctrine, and thus could be grounds for a claim of infringement. You should be very very careful if you are thinking about incorporating or using another companies registred or even unregistered mark in to the name of your product (personally, I wouldn't do it, too much risk).

Again, I'm not a lawyer, this isn't legal advice, and the above is based on my own personal experience. This is a very complex area of law and it's best to seek formal legal advice in the naming of your product, especially if you are considering incorporating another companies registered trademark. For formal legal advice on Intellectual Property, I recommend Vic Lin at ICapLaw; incidentally he has a great blog with loads of information on this topic which you can find here.

Obvious disclaimer: I'm not a lawyer. I haven't even spoken to a lawyer in a few years. Even if I was a lawyer, I wouldn't be your lawyer. If there's any possibility of a lawsuit, keep in mind that "dude on the Internet said so" is never admissible as evidence.

That said, the short answer to the "headline" question is that you need to know what jurisdictions you care about (places you're likely to be doing business, whatever that means for your situation) and search the trademark office's records there. For a local company, the registration might only be with the city.

However, common (un-registered) trademarks can also be protected, and can seem stronger. They're not limited to the government where they were registered, but are usually considered to be in effect wherever people are likely to connect the name with a product or service. And whereas you need to defend your registered trademark to keep it, you can sit on a common trademark and decide when you care enough to sue over infringement.

In the end, you can also always just ask the maintainers (without giving any specifics, so that they don't use your situation to invent policy that doesn't yet exist) what their trademark policy is or if they have a media kit. That usually gives the whole story of what they consider acceptable and how they'll respond if you don't comply.

The simple rule is to best to not use someone else's brand/name (especially given they're an established business) in yours. It's less headache for all.