You should register and insure your second car in the state where it is primarily driven. Your state of residence isn’t as important to car insurance companies as the address where your car is physically located.
The location of your vehicle is a rating factor for car insurance companies. Your rates for your two cars will be different because their different locations have different risks.
Your insurance requirements may be different, too.
For example, if your primary car is located in Ohio, the state minimum requirements are 25/50/25 (meaning $25,000 per person and $50,000 per accident for bodily injury liability and $25,000 for property damage liability). If your secondary car is located at a vacation home in Florida, the state only requires property damage liability coverage of $10,000 and, because it’s a no-fault state, personal injury protection of $10,000.
To protect your assets it’s best to carry more liability coverage than the state requires, such as 100/300/50. And if you want your car to have coverage for damage it may receive, you need physical damage coverages of collision and comprehensive on each car in each state.