Unique universe was not the hope

Physicist Alan P. Lightman writes in the current Harper's magazine:

An example of a fundamental principle in physics, first proposed by Galileo in 1632 and extended by Einstein in 1905, is the following: All observers traveling at constant velocity relative to one another should witness identical laws of nature. From this principle, Einstein derived his theory of special relativity. An example of a fundamental parameter is the mass of an electron, considered one of the two dozen or so “elementary” particles of nature. As far as physicists are concerned, the fewer the fundamental principles and parameters, the better. The underlying hope and belief of this enterprise has always been that these basic principles are so restrictive that only one, self-consistent universe is possible, like a crossword puzzle with only one solution. That one universe would be, of course, the universe we live in. Theoretical physicists are Platonists.

No, the underlying hope was not for a unique self-consistent universe. Here is Galileo's ship from 1632:

Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals. ... With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. ... When you have observed all these things carefully (though doubtless when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still.

This is an argument against uniqueness. He was arguing that we would not notice the motion of the Earth, and this only implies that geocentric and heliocentric models are indistinguishable.

Here is Poincare's 1900 relativity paper, where he addresses his previous criticisms of Lorentz's relativity theory:

t would no doubt seem strange that in a monument raised to the glory of Lorentz I would review the considerations which I presented previously as an objection to his theory. I could say that the pages which follow are rather in the nature of an attenuation rather than a magnification of that objection.

But I disdain that excuse, because I have one which is 100 times better: Good theories are flexible. Those which have a rigid form and which can not change that form without collapsing really have too little vitality. But if a theory is solid, then it can be cast in diverse forms, it resists all attacks, and its essential meaning remains unaffected. That's what I discussed at the last Congress of Physics.

Good theories can respond to all objections. Specious arguments have no effect on them, and they also triumph over all serious objections. However, in triumphing they may be transformed.

The objections to them, therefore, far from annihilating them, actually serve them, since they allow such theories to develop all the virtues which were latent in them. The theory of Lorentz is one such, and that is the only excuse which I will invoke. ...

It appears that the principle of relativity of motion, which is not clearly true a priori, is verified a posteriori and that the principle of reaction should follow. ... It is the case that, in reality, that which we call the principle of relativity of motion has been verified only imperfectly, as shown by the theory of Lorentz.

Einstein's famous 1905 relativity paper called Poincare's relativity principle a "postulate", but that was not how relativity was discovered. Poincare says that it is "clearly" not a postulate, and was not motivated by a search for a unique theory.