It's an ether. A halogenated ether, not an alkane, nor an alkyl halide though it has halogens on it.
Surgical anesthesia from the beginning has been accomplished with diethyl ether, which works sufficiently well. While it disrupts neuronal permeability to the point of unconsciousness, it has a specific danger. 1) it is highly volatile (low vapor pressure) 2) it is extremely flammable.
However, the use of halogenated ethers reduces the risk of explosion and is still used today. The halogens on these compounds are not very reactive and do not react like typical alkyl halides. If they did, they would be extremely corrosive. The groups do not leave readily since 1) this is the gas phase we're talking about, not protic solvent 2) loss of a leaving group leads to an inductively destabilized charge.
In terms of function (versus structure--not versus functional group identity) they are ethers. If you wish to talk about functionality in terms of reactivity, fluoroalkanes do not behave like most alkyl halides do. However, if you want to just identify the functional groups the two mentioned are the only possibilities.
There are far more important properties of these compounds than the functional groups they contain which are among the simplest.
These compounds are used because they are relatively unreactive. Ethers are common solvents and don't react except with concentrated acid. The halides are not good leaving groups (no protic solvent) so they don't react either. Think of methylene chloride. Most organic solvents are flammable. CH2Cl2 is not. Nor does it react even in the presence of Grignard reagents. The better question is why do you think that is?