In my textbook it is said that they are soluble in non-polar solvents (which makes sense), but also soluble in polar solvents (halogenoalkanes and some ethers like ethoxyethane) It is said that the polar solvents can induce a dipole in the fatty acid chain. If this is the case, why does water not induce a dipole in the chain?
Polar is not a black-and-white distinction. There are weakly polar compounds (like ether), medium polar compounds (dichloromethane) and strongly polar compounds (dimethyl sulfoxide). Fats (i.e. triglycerides) can dissolve in weakly and maybe moderately polar solvents but not in strongly polar ones.
While a dipole can be induced in triglycerides to some extend, most of the long nonpolar side chain can only give weak induced dipoles. Thus, weakly polar solvents may still dissolve these by dipole–induced dipole interactions but the triglycerides are just too nonpolar for strongly polar solvents.
Water is one of the most polar solvents and additionally able to hydrogen bond. While triglycerides can accept hydrogen bonds around the glycerin moiety, they are just not polarisable enough for water to act. Hence, water prefers to form its own phase.
I think it depends on the number od carbon atoms in fatty acid chain. For example butyric acid is a carboxylic acid, like fatty acids, and it is partially soluble in water. All fats and oils are partially soluble in cold methanol, ethanol, acetone, but soluble when they are heated, so there is also this effect.
Also, it depends what do you think is polar or non-polar enough, for example you mentioned ether. Ether is considered non-polar rather than polar, but yet it is more polar than hexane or pentane for example.
And yes, polar solvents can induce dipoles in non-polar compounds, that is dipole-induced dipole interaction, type of van der Waals forces. But those interactions are not favorable in the case of water for example, since water is "trapped", molecules of water would rather build hydrogen bonds instead of dipole-induced dipole interactions. That is why water escapes from nonpolar molecules and nonpolar molecules build micelles. Some compounds and solvents are emulsifiers so they distribute micelles, and prevent them to form emulsions, so it seems like fats remained soluble.