Do I automatically get a UAE residence visa when I register a trade license?

No, the license grants you eligibility to apply for visas, but they are not issued automatically. The number of visas you can obtain depends on your jurisdiction and facility:

Mainland: The visa quota is generally linked to the size of your office (typically one visa per 9 sq. m). 
Free Zone: Visas are usually allocated based on the license package you purchase (e.g., zero-visa, 1-visa, or 3-visa packages).