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Suppose we need to predict a real number in fixed range, for example, [0 .. 5], and our Y can be 3.14, 2.4654 etc. What is the name of this kind of tasks (to be able to search further) and what are the approaches to solve this problem?

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This is a regression problem with a "limited dependent variable".

One very common approach is to use a sigmoid transformation as the final step in your model. For example, the logistic transform $f(x) = \frac{5}{1 + e^{-x}}$ is constrained to the interval [0, 5].

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