Well, to start with, there are many examples of creative naming of music centuries before Beethoven. Renaissance music has many pieces with very evocative names or nicknames. For example the pieces of English composer John Dowland ( 1563-1626) include “Come, Heavy Sleep “ , “Flow, my Tears” , “ I Saw my Lady Weepe” , and Lachrimae, or Seven Teares, Figured in Seaven Passionate Pavans. Before that, Medieval music was also given names in a similar way. I am not sure about the ancients. The ancient Indians, Chinese, Egyptians, Greeks and Romans as well as other cultures all had rich musical traditions , but I am unaware of any actual surviving songs or pieces from those civilizations. However it is a reasonable conclusion that they probably named their songs.

To address the second question ( Why do most Classical music pieces just have “bland” names ) I would like to use an analogy for a moment. Think about how species are named in science. They have a specific format, in order to catalogue them and make it easy to specifically name them. But they usually also have a “nickname “ or common name by which they are usually referred in casual conversation.

It’s similar with Classical music. Pieces are given a name according to a specific format developed in order to keep track of them more easily. The format includes the type of piece, the number in a chronological list of that same type of piece that the composer wrote, and either an opus (published work) or catalogue number. Let me link a previous comment here where I explained the naming system in detail. Not all pieces have a nickname, but many do. These nicknames were either bestowed by the composer or later critics. As a special note, sung music , no matter the type or era, is usually given a name based simply on its lyrics.

Taking the specific example of Beethoven’s Moonlight Sonata: Its full formal name, which was given to it when it was published in 1801, is Piano Sonata No. 14 in C♯ minor "Quasi una fantasia"( “in the style of a Fantasia) , Op. 27., no. 2. It acquired its nickname in 1832 when the German music critic Ludwig Rellstab compared the sound of the 1rst movement ( section) to moonlight. Some critics said the name was inappropriate, that the 1rst movement sounded more like a funeral march than moonlight, but nevertheless the name stuck and now the Piano Sonata no. 14 is known universally as simply “Moonlight Sonata” .

u/Bach09 makes some good points, but I'd also like to link to a previous answer I've written to this question that delves a bit into why you are more likely to see pieces given names by their composers during/after the 19th century than in the 18th century.