The United States Army's Institute of Heraldry says "DoD component names, insignia, seals, emblems, symbols, logos, and other marks and designs are protected by law from unauthorized use. Information regarding trademark..." So, trademarking elements of heraldry in the United States is an official act.
Company logos are often coat of arms. Colors, sounds, motions, scents, etc. can be trademarked. A sound that is trademarked is still a sound. Arms that are trademarked according to this logic are still arms.
In countries that officially regulate private arms it is common for a company with a patent of arms to apply for a trademark.
Reasons to trademark a personal coat of arms include getting official recognition, making the arms inheritable, having an official government record of it, etc. Protecting the coat of arms, per se, is actually low on the list because the odds of someone actually copying your obscure arms is near zero, especially since personal arms are usually overly complex these days.
The reason to protect your arms is not protection per se, but the use of protection is to give it official recognition. It will be registered by a government entity, and potentially the record will be forever, even if you do not pay your fee at the term's end. Legally, your arms will be yours beyond mere copyright.
As you, the reader, is likely a hobbyist, you should not go overboard and spend much money on this. Register the trademark with your state. At the time of writing, some states charge as little as $10 to register and often last 10 years. You do not even have to register in your state; register online in a low-cost state. If you register out of state to save money, claim a common law trademark in your state. Remember, for the trademark to be valid you would need to do some type of commerce in the state where it is registered.
A trademark can only be used for commerce. Being a philanthropist, as in being a self-funded charity that does not solicit funds, is sufficient to be considered being in commerce. Specimens of your trademark may need to be submitted, so put your arms on your checks. Buy a custom magnet and create letterhead. When making a charitable contribution, use your checks with your coat of arms. If you are already making charitable contributions, then it will not even cost you extra. Claim the donations under your trademark. Learn about how to submit a trademark drawing. Consider submitting a black and white drawing. Education is another category of trademark.
The drawback is the periodic registration and donation obligation, unlike Britain, where coats of arms are forever. However, in the olden days a bearer of coats of arms did have an ongoing obligation: to fight when called by the king. Paying money to charitable causes brings relevance to a coat of arms. It would be an oblation such as knights make to their knighthoods. Oblations and education are commerce that will bring respect to your coat of arms.
Trademarking also provides another status: reputational harm caused by false statements. If someone libels by using the arms as an identifier, something would arise due to reputation harm as in product disparagement. Your trademarked arms is a source of goods, for example your donations. If someone lies with malice and causes an armiger damages by saying that the source of the trademarked arms actually did not make the donations then it is likely that there would be standing for a lawsuit.
Trademarks were started by craftsmen that marked their products. Trademark law started when the craftsmen needed to protect the goodwill they built up after years of hard work. So, with any goodwill you build to your mark is considered property. Thus, with a trademark, your coat of arms becomes legally associated with you as a person. Libel against your coat of arms becomes libel against you. Trademark is a merchant/bourgeois concept, as opposed to a coat of arms, which is a military/gentry/noble concept. The two come from different lineages.
Trademarking for the American arms' bearer makes the coat of arms legally his own in perpetuity, something a copyright does not do. With a trademark the coat of arms gains a higher legal status.
Every state and territory law is different, so all provided information is general. Perhaps get an inexpensive legal consultation with an attorney through a legal plan or a bar referral.
Gaining limited legal status for your arms is very inexpensive and easy.
