My son was born on October 2nd of 2019 and I was wondering why I couldn't claim him on my taxes for that year

You have to say he  lived with you for  the full year.

The child may be claimed as a dependent if you meet all of the tests for either a qualifying child or a qualifying relative.

IRS publication 501 page 11 outlines the requirements.

A child born anytime during the tax year will qualify you for the full $2,000 Child Tax Credit.  Be sure you entered the birthdate correctly AND indicated that he lived with you for 12 months. Even though he was born late in the year, he is treated as if he lived with you all year.