The complaint email from Amazon or Etsy says "intellectual property". That is a category, not a claim. Inside it are three different rights, and the notice tells you which one if you know where to look.
Trademark: names, logos, slogans
A trademark complaint cites a word, a logo or a phrase, usually with a registration number. It is about the name on your listing, your product or your packaging. This is the most common complaint sellers receive, and the only one of the three you can screen for in advance, because trademarks sit in public registers.
Response: look the mark up. If it is live in your class in that country and your listing uses it, remove the use and say so. If it is dead, in unrelated goods, or the complaint cites a name your listing does not contain, reply with the register printout. Rights owners send bulk complaints and a fair share of them are wrong.
Copyright: images, text, artwork
A copyright complaint is about creative work: product photos you took from the brand's site, description text copied from another listing, artwork or a character on your design. No registration number is needed and none is usually cited.
Response: if you copied it, replace it with your own. If the work is yours, say so with proof (your original files, dates). A trademark-style counter-notice about the name does nothing here because the name was never the issue.
Patent: how the product works or looks
A utility patent complaint says your product uses a protected invention. A design patent complaint says it looks like a protected design. These come with a patent number and, on Amazon, sometimes through the neutral patent evaluation process.
Response: this is the one where sellers should not improvise. Removing the listing stops the bleeding; deciding whether the patent actually reads on your product needs someone who has read the claims. Do not send a trademark search result as a defence to a patent claim. It signals you did not understand the notice.
Why the mix-up costs accounts
Sellers who answer a copyright notice with "I do not use their trademark" get marked as non-responsive. Sellers who answer a trademark notice by changing the product photos leave the name in place and get a second strike. Read the notice for the word trademark, copyright or patent, then respond to that one thing.
For the trademark case, the search above gives you the facts in seconds: who owns the mark, whether it is live, and whether your goods overlap. Those three answers decide the response.