Is College Algebra the same thing as Linear Algebra?

I'd say neither.

College Algebra is a fast paced refresher of algebraic concepts which may touch on what was covered in both Algebra I and Algebra II. The high school courses are taught with 180 days of instruction while college algebra is usually a semester of about 30 hours of class instruction (and the rest for your self-study).

Linear Algebra is an extension of the study of linear equations (aka lines) but getting into matrices and vector spaces. It may touch a little on geometry such as projections of vectors onto other vectors, cross products, transformations, etc.

If you are feeling like you need to refresh your understanding of Algebra from high school, College Algebra is worthwhile. If you are getting into other areas of mathematics, engineering, computer graphics, etc. then Linear Algebra is probably a better course.

If you are just looking to satisfy a mathematics requirement but don't really want to learn anything new, there are other options like math for non-majors.

No it is not the Same.   College Algebra is advanced Algebra and can include Linear Algebra and all other categories of Algebra.

College algebra is about learning the basics of elementary algebra. Like the basic rules for simplifying and using algebraic expressions. Linear algebra is the study of vector spaces. In particular, a focus on the properties and usage of matrices for dealing with problems involving systems of linear equations.

College Algebra has a lot of overlaps with Algebra II, although College Algebra will do additional topics.

Linear Algebra is something math and engineering majors take.