I have 2 residences one in TX and the other in North eastern state. Our primary residence is in Texas where we spend more than 6 months. The NE state has asked us why we are not filing taxes in that state. We told them that we live in TX. What documents are needed to prove that? We have Driver's Licenses and Property Tax documents. I am sure thousands of people do this.
You can easily prove your residence by showing power bills, water bills, etc with your name on it, but it might make no difference:
Many states require you to file and pay taxes if you have income while you are in the state, starting from 30 days in the year. This applies to business travellers even.
As a consequence, if you spend more than 30 days there, you might owe taxes there. Worst case, if you are on the road a lot, you could owe state tax in up to 12 states (for the same income!), unless they specifically have agreements to recognize already paid taxes. Welcome to independent state tax laws.
First a couple of items.
If a state knows about you, they will be interested in connecting your income to their state. If they can establish that you are a resident of their state they may even be able claim income you earn in another state. Many times two neighboring states will have a reciprocity agreement, where you only have to pay income taxes where you live. But if there is no reciprocity agreement they will give you a credit for the income you pay tax you pay to the other state. For you unidentified state X and Texas are involved. The unidentified state will be more than happy to credit you the $0 income taxes you will pay to Texas and then collect the rest.
You will have to contact the unidentified state to know what proof they need. The more that you have that point to Texas the better off you will be. Drivers license, voter registration, and the fact that all your bills are sent to Texas helps. Property tax bills aren't enough because you can own property in multiple states, but live in a place you rent.
It sounds like this is a question for a CPA or an enrolled agent. It is not a question for the internet. State tax codes have varying understanding of what is "residence." You can be a resident, for tax purposes, in a state where you own no land, lack a driver's license, and are only present for work due to a long but temporary contract. Likewise, you could owe tax on income earned in another state. For that matter, under the current version of the US tax code, your "tax home," may not be where your home is.
I wouldn't respond to the state making the request until I had spoken with a CPA, enrolled agent or tax attorney. I can think of one state that calculates taxes due on multiple factors. It does NOT consider residency in the legal sense that you are thinking about it; it just provides a non-resident tax form. You could add taxes to yourself by being a non-resident. Go visit a CPA.