Does Texas liability insurance cover if the two cars involved in the accident are register under the same persons name?

Answer 1

Here is the problem with that,

"It is not legally possible to be liable to ones self"

So no, liability only coverage will not cover damage to your own property, That's what the comprehensive and collision coverage is offered to you for when you purchase your auto insurance. Comp and Collision provides coverage for your own property, Liability provides coverage for property and injury of others.

If you bought liability only, there is no coverage for your own property.

Answer:

It should, and if it doesn't, there should be other parts of the policy that can pay out. Coverages such as uninsured motorists or medical payments.