Texas licenses plumbers and electricians. It does not license general handymen at all. That single fact explains most of the confusion, and most of the bad advice you will get.
What Texas does not license
There is no state handyman license in Texas. No exam, no registration, no number to check. Anyone can print a card. That is not a loophole being exploited, it is simply how the statute is written, and it means the phrase "licensed handyman" is close to meaningless on its own.
What you can check is insurance. General liability cover is what protects you if something is damaged, and it is a fair thing to ask for before anyone starts. Ask for the carrier and the policy period, not just a yes.
What Texas does license
Plumbing is regulated by the Texas State Board of Plumbing Examiners. Electrical work is regulated by the Texas Department of Licensing and Regulation. Both require a real license, both are checkable, and neither is optional.
In practice the line falls in a fairly predictable place. Swapping a faucet, a trap, a garbage disposal or a shut off valve is fixture work. Moving a supply line, altering the drain stack or touching the water heater gas connection is plumbing. Replacing a light fitting on an existing circuit is fixture work. Adding a circuit, altering the panel or running new cable is electrical.
This is why our plumbing work stops at fixtures and valves, and why anything past that line goes to a licensed plumber rather than being quietly done anyway.
Permits are a separate question
A license is about who may do the work. A permit is about whether the work has been inspected. They are not the same thing and you can need one without the other.
Most of a bathroom remodel or a kitchen remodel that stays inside the existing footprint will not need a permit. Move a wall, relocate plumbing, or change the electrical layout and it will.
Here is the part people outside San Antonio get wrong: several cities inside our service radius issue their own permits, entirely separately from the City of San Antonio. Kirby, Castle Hills and Alamo Heights each run their own permit office. Same metro, different process, different timeline. A contractor who only ever works one side of town will sometimes not know that.
The honest version
A good handyman turns work down. Not because the rules are complicated, but because the jobs past the line are jobs a specialist will do faster, cheaper and with a warranty that means something.
If someone tells you they can do absolutely anything, that is not confidence. Ask what they will not do. The answer tells you more than the list of what they will.
Need this doing rather than reading about it?
Uncle Sam Handyman LLC covers a 35 mile radius at one rate, with a free written estimate before any work starts.