Is a relational database just a $ \geq 2$-dimensional array?
No, it isn't. A data structure can be described in two ways: 


How does it store the data.
Which operations are supported, their performance (usually given as asymptotic time- and/or space-complexity), and what laws do they obey.


Obviously, a database doesn't store all its data as a single array.

Arrays support, as basic operations, retrieving an element by index and storing an element under an index. Operations required for databases are (to a first approximation) given by relational algebra: selection, projection, union, difference, and join (or product).