It can, and does, but it depends on the exact nature of the accompaniment, and many times the best terminology is exactly what you've used already ... "accompaniment".
For example, there's counterpoint, which Wikipedia currently defines as "the relationship between voices that are harmonically interdependent (polyphony) yet independent in rhythm and contour". This generally means there are two melodies going on at the same time, complementary to each other. Compare the term "countermelody", which is related.
Here's an example that appears to be a transcription of a two-part invention by J.S. Bach:
This is counterpoint; you can see that the lower/"left hand" voice plays something similar to the upper/"right hand" voice just a couple of beats later than the upper voice ... almost an "echo".
But counterpoint is generally considered to be a classical technique, although you could, I suppose, argue that it's still used today when "trading licks", or especially by background vocalists. Compare to this also the "call & response" concept which came to us from African music via "Negro Spirituals" and slave work songs in the early USA.
Much more often than "counterpoint" (or polyphony) we will hear a "chordal" accompaniment, which might have several words to describe it, as well. Most often, melody will move or change pitches more frequently than the accompaniment, in which case the music is said to have a "slower harmonic rhythm".
Here is a piece of music, easily playable on piano, that would demonstrate "chordal accompaniment". Note that the pianist's left hand would play half note chords (Gm and F) while the right plays the melody:
Of course, this is all really a simplified set of examples. The accompaniment for this last example could be played in an arpeggiated style, or with the "Alberti bass", or any number of variations ... and that's just on the piano.
Add in the complexity of a band or orchestra, and you can get really complex pretty quickly.
Here's a transcription of the Beatles opening to "All My Lovin'":
Perhaps you can see Paul on bass guitar, playing a "walking" bass line of all quarter notes (on each beat), Ringo playing a "swing rock" pattern, and the 2nd guitarist (I'll presume it was George) playing triplet chords (that is, 3 strummed chords on each beat of the song, or strumming three times faster than Paul was).
And here's the opening page of the last movement (not the more famous first movement) of Beethoven's 5th symphony:
Perhaps you can observe "chordal accompaniment", but with winds playing long (half) notes, high strings playing staccato notes, and lower strings playing a repeating pulse of eighth notes.
What you've really discovered here is often called "texture", and there are, usually, 3 musical textures:
Monophony (that is one instrument/voice producing one line of music; one note at a time).
Homophony (a melodic line of music accompanied by multiple notes of lesser importance than the melody).
Polyphony (of which counterpoint is the major example), two or more simultaneously melodic structures occurring more or less simultaneously.