Not very similar. The problem with ether is that it fixed a preferred frame. So some objects were truly still and some objects were truly moving.

The Higgs field also permeates all objects, but the mathematical structure is different and it has no preferred frame, so there is no truly still object, all the movements are relative.

This is not a strange thing. The photons are the bosons of the electromagnetic field, that also permeates all objects and it has no preferred frame, so there is no truly still object, all the movements are relative.

And the gluons are the bosons of the strong force field with exactly the same properties.

Even the electrons have and associated field that that also permeates all objects and it has no preferred frame, so there is no truly still object, all the movements are relative. But electrons are fermions, no bosons, so some properties are different, but they have an associated field.

The same happens with the up and down quarks (and neutrinos). Each one has a associated field, with the same properties. (And the other particles too, but this is becoming too repetitive.)

The strange thing is not that the Higgs Bosons have an associated field. The strange thing is that the vacuum expectation value of the Higgs Fields is not zero.

In all the other cases (photons, electrons, ...) the vacuum expectation value is zero, so if you have an empty box, there are almost no particles there (some virtual particles appear, but in some sense they are only a few, (in another sense they are a lot, but it's better to think that they are only a few).) The important thing is that if any particle pass through the box, it will almost not bounce against any other particle (photons, electrons, ...) because it is empty.

But for the Higgs fields the vacuum expectation value is not zero! So even inside the empty box the value of the Higgs field is not zero, there is something like a background value. If any particle passes through the box, it will bounce many many many times against the background even if the box is empty. An easy way to interpret all this bouncing is to say that the particle has apparent mass.

But the Higgs field is not even, it has bumps, and those bumps are the Higgs Bosons. The empty box has only very few Higgs Bosons (or a lot in another technical sense). The normal particles can bounce against the background (vacuum expectation value) and we call it "mass" and they can also bounce against the bumps (bosons) and we call it "interactions".