How can I stop seeing the migration cape on other players?

Capes can be removed by modifying the game's Core Shaders via a resourcepack (possible 1.17 and onwards).

The rendertype_entity_solid.fsh shader controls the cape rendering. The cape can be made entirely transparent by inserting this bit of code:

if (hasCape){
    discard;
}

Like so:

#version 150

#moj_import <fog.glsl>

uniform sampler2D Sampler0;

uniform vec4 ColorModulator;
uniform float FogStart;
uniform float FogEnd;
uniform vec4 FogColor;

in float vertexDistance;
in vec4 vertexColor;
in vec4 lightMapColor;
in vec4 overlayColor;
in vec2 texCoord0;
in vec4 normal;

out vec4 fragColor;

void main() {
    
    vec4 color = texture(Sampler0, texCoord0) * vertexColor * ColorModulator;
    bool hasCape = textureSize(Sampler0, 0) == ivec2(64,32);
    if (hasCape){
        discard;
    }
    color.rgb = mix(overlayColor.rgb, color.rgb, overlayColor.a);
    color *= lightMapColor;
    fragColor = linear_fog(color, vertexDistance, FogStart, FogEnd, FogColor);
}

The rendertype_armor_cutout_no_cull.vsh shader controls armor rendering, including elytra. The cape texture can be removed from the elytra by inserting the following code by McTsts:

    // removes cape elytras
    vec2 texSize = textureSize(Sampler0,0);
    if( texSize.x == 64 && texSize.y == 32 && // check if size matches a cape
            texture(Sampler0, vec2(0.0, 1.0/32.0)).a == 1.0 && texture(Sampler0, vec2(12.0/64.0, 16.0/32.0)).a == 1.0 && // check if skin transparent pixels have color
            (   // support both types of cape texture
                (texture(Sampler0, vec2(0.0, 18.0/32.0)) == vec4(1.0) && texture(Sampler0, vec2(0.99, 0.99)) == vec4(1.0) && texture(Sampler0, vec2(0.99, 0.0)) == vec4(1.0)) // for capes filled with white, check for white pixels
                || (texture(Sampler0, vec2(25.0/64.0, 25.0/32.0)).a == 0.0 || texture(Sampler0, vec2(30.0/64.0, 10.0/32.0)).a == 0.0 || texture(Sampler0, vec2(9.0/64.0, 25.0/32.0)).a == 0.0)  // otherwise check of transparent pixels
            )
    ) gl_Position = vec4(0.0);

Like so:

#version 150

#moj_import <light.glsl>

in vec3 Position;
in vec4 Color;
in vec2 UV0;
in vec2 UV1;
in ivec2 UV2;
in vec3 Normal;

uniform sampler2D Sampler0;
uniform sampler2D Sampler2;

uniform mat4 ModelViewMat;
uniform mat4 ProjMat;

uniform vec3 Light0_Direction;
uniform vec3 Light1_Direction;

out float vertexDistance;
out vec4 vertexColor;
out vec2 texCoord0;
out vec2 texCoord1;
out vec4 normal;

void main() {
    gl_Position = ProjMat * ModelViewMat * vec4(Position, 1.0);

    vertexDistance = length((ModelViewMat * vec4(Position, 1.0)).xyz);
    vertexColor = minecraft_mix_light(Light0_Direction, Light1_Direction, Normal, Color) * texelFetch(Sampler2, UV2 / 16, 0);
    texCoord0 = UV0;
    texCoord1 = UV1;
    normal = ProjMat * ModelViewMat * vec4(Normal, 0.0);
    
    // removes cape elytras
    vec2 texSize = textureSize(Sampler0,0);
    if( texSize.x == 64 && texSize.y == 32 && // check if size matches a cape
            texture(Sampler0, vec2(0.0, 1.0/32.0)).a == 1.0 && texture(Sampler0, vec2(12.0/64.0, 16.0/32.0)).a == 1.0 && // check if skin transparent pixels have color
            (   // support both types of cape texture
                (texture(Sampler0, vec2(0.0, 18.0/32.0)) == vec4(1.0) && texture(Sampler0, vec2(0.99, 0.99)) == vec4(1.0) && texture(Sampler0, vec2(0.99, 0.0)) == vec4(1.0)) // for capes filled with white, check for white pixels
                || (texture(Sampler0, vec2(25.0/64.0, 25.0/32.0)).a == 0.0 || texture(Sampler0, vec2(30.0/64.0, 10.0/32.0)).a == 0.0 || texture(Sampler0, vec2(9.0/64.0, 25.0/32.0)).a == 0.0)  // otherwise check of transparent pixels
            )
    ) gl_Position = vec4(0.0);
}

Resourcepack download

Unfortunately this doesn't seem to be possible as of right now. Upon some research, I can't find any mods to do it and, since migrator capes are given officially by Mojang (and so they display on any client), I highly doubt there will ever be any official way either. However, making a mod to do this is likely pretty easy, so it's possible that somebody will eventually make one that'll do what you want.

Apologies if this doesn't really answer your question, but there isn't really much else to say.