Sir John Packington was a member of the court of Queen Elizabeth in the 1500s. Although we have no information on what kind of person he was, he is remembered today for an act of rudeness. He built a dam across a stream on his property. The pond behind the dam flooded his neighbor's field, who took the case to court. Packington argued that, since the dam was entirely on his own property, he had every right to build it. He felt that he could do anything he wanted to do on his own property. The court disagreed, as did the court of public opinion. A satirical song, now lost, was written poking fun at him. The tune for that song became the tune for at least 100 songs that we know of, more than even Greensleeves.
To understand why the words of some songs go to certain tunes, you have to understand poetic meter. Each line of a poem that is not written in free verse has a number of feet. A foot consists of several syllables. The most common type of metrical foot is called an "iamb", which consists of an unaccented syllable followed by an accented syllable. Another type of foot is an anapest. An anapest consists of two unaccented syllables followed by an accented syllable.
One verse of The Star-Spangled Banner consists 8 lines of anapestic tetrameter, but with a few twists. Tetrameter means that each line has 4 feet (tetra-meter, get it?). Poems with verses that consist of 4 lines of anapestic tetrameter are fairly common. For example, this is the meter of The Night Before Christmas.
Some of the lines of The Star-Spangled Banner have feminine endings. This means that an extra unaccented syllable is added onto the end of the line, and this becomes part of the required rhyme. That makes it unusual. In addition, the fifth line is broken into two lines. Think about the sound of "And the rockets red glare, the bombs bursting in air." That means that no tune meant for 4 lines of ordinary anapestic tetrameter can be used.
Is it an accident that Anacreon uses the exact same meter as Packington's Pound? Maybe. Since we don't know whether the words for Anacreon were written at the same time as the tune, it is at least possible that Packington's Pound was the tune that Anacreon was sung to originally.