If I follow you correctly, someone resident in NY has created ABC, LLC. You would like to create ABC, Inc. in California. With regard to the corporation, it depends on the guidelines of the Secretary of State. Some jurisdictions will let you register ABC LLC and ABC, Inc. owned by different persons. Other states, e.g., NY, will not allow it. You should go to the California Secretary of State website and search for names to see if ABC, Inc. is available.
As to the trademark, if you use the mark in commerce, you can register the mark with the USPTO. The standard is reasonable likelihood of confusion. If your mark associated with ABC, Inc. is not likely to confuse customers of ABC, LLC, you should be fine, assuming no one else in the country is using ABC, LLC or ABC, Inc. in a similar industry.
Based on the names alone it is not a problem to incorporate in California, but that is the beginning of the analysis. Are you in the same business or different businesses. Does the NY businees have a website like ABC.com? If the same business and NY has been selling the product in interstate commerce before you, then you cannot register the trademark. There is always the possiblity of a lawsuit in California for unfair competition to obtain an injunction. If you are in totally different businesses, this trademake issue may not be absolute depending on how similar the products are (jewelry and apparel vs. jewelry and automobiles.) So much depends on the facts. My bottom line is that almost no name is worth it if the facts make it a close call. Use your creativity instead.
The above is general legal and business analysis. It is not "legal advise" but analysis, and different lawyers may analyse this matter differently, especially if there are additional facts not reflected in the question. I am not your attorney until retained by a written retainer agreement signed by both of us. I am only licensed in California. See also avvo.com terms and conditions item 9, incorporated as if it was reprinted here.
The above answers are correct. But let me muddy the waters a little more. Name protection is a really complicated area of business law. Perhaps complicated is the wrong word - it's simple in that "whoever uses a name first in a particular marketplace has superior rights to anyone using a confusingly similar name later."
But how that applies in practice, and especially in light of the Internet and "a particular marketplace" being much broader these days, is complicated. I strongly recommend you talk to an attorney because while the California Secretary of State only cares about their own list and will happily register a corporate even if that name is registered in another state, there are implications beyond whether you can register the corporate name.
Likewise, registration of a trademark is not legally necessary, but the analysis of whether you're going to be a late comer with a confusingly similar name in a marketplace is an important analysis, and to do it well an attorney would need more information than you want to post publicly.
You don't tug on Superman's cape; you don't spit into the wind; you don't pull the mask off the old Lone Ranger; And you don't get legal advice from a free Q&A page on the Internet. The above is a general statement of the law or just my opinion. I am not saying whether it applies to your situation or not because I don't know the details and youâ€™ve not hired me as your attorney.