is Morley-Michelson like Doppler?

[Svartalf]

Hello... You may already know that I totally suck at maths and have only a very basi science education.
I just have a thing I'd like help with from the better educated.

So I'm watching a little vulgarisation series about the history of Euclid, mathematics, the 5th postulate, and the birth of non euclidean geometry. Except now it has diverted about how we discovered that there is no such thing as the aether.

So, as I understand it, the Morley Michelson experiment aimed at checking the influence of the movement of earth on the speed of light.

For whatever reason, this strikes me as the light version of the Doppler-Fizzeau effect (you know, how the movement of a moving object toward or away from you affects the sound you hear). Now, I'd like to know if my intuition, however vague is basically correct, of if I'm setting up an analogy that's just shitty.

[pErvinalia]

The light version of the Doppler effect is red/blue shift. The speed of light is constant, though, regardless of the shift.

[Svartalf]

I walked straight into that one didn't I?