It’s certainly legal to have your parent and child company have the same name (assuming they are located in different states). Let’s suppose you wanted the name ACME, LLC (and it was available in both states), then you could have ACME, LLC in both states. The reason for this, is technically, the legal reference for your company name is “ACME, LLC, a Florida limited liability company” (Assuming this ACME, LLC, is located in Florida).
We do not recommend you do this, however, because most folks don’t remember to type out the entire “ACME, LLC, a Florida limited liability company”. Instead, most folks like to refer to just ACME, LLC. The problem is, if you have two companies named ACME, LLC, which company are you referencing in a contract, when you have ACME, LLC, instead of “ACME, LLC, a Florida limited liability company?” What happens, is you create am ambiguity in the contract that can be potentially leveraged by the other party. This isn’t a problem, until it is a problem. If someone wanted to sue you and/or ACME, LLC, you’ve basically given them an opening to pick which ACME, LLC, they want to sue and/or they could sue both. They could also argue you were trying to be intentionally vague and fraudulent, and then try to use that excuse to “pierce the corporate veil.”
While that is a poor rationale to pierce the corporate veil, it is nonetheless something available to try and you’re left trying to defend yourself against such a claim.
It’s best to not use the same name for parent/child companies. Instead, given the child company is supposed to be your Operating Company, name the child as appropriate for branding purposes. Then, the parent can be named whatever you want, and if you want to use the same name as the child, add the state to the name. For example, if the child was ACME, LLC, a Florida limited liability company, and the parent was located in New Mexico, then I would recommend naming the parent ACME NM, LLC, or ACME Holdings, LLC, or something that can help distinguish it from the Florida company.
Note that it is not acceptable to simply drop the “LLC,” since it or something similar (i.e. “ltd.,” or “L.L.C.” or something similar to indicate it is indeed a LLC) is required as part of the legal name of the company.