I would like to model my problem using something similar to Item Response Theory, but my responses are not binary. They are continuous in $[0, 1]$.
What are these models/the research field called?
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The accepted answer does not give models where the response is bounded between [0,1]. There are IRT models for exactly the case where the response variable is continuous, but bounded in this way.
For example, Samejima  describes exactly this case. These models are sometimes called Continuous Response Models (CRM), and the case they're addressing is sometimes called Continuous Response Format (CRF). They are the standard factor analysis setup, but add a transformation to bound the responses within an interval (in the same sense that the standard IRT takes factor analysis, and adds a link so responses will be, say, 0 or 1).
Ferrando  has a good paper on this, and the R library estCRM looks like it can implement it (though I haven't used it!).
I agree that the accepted answer does not point towards specifically dedicated models. See for instance: