Whilst there might be some rechargeable tail lamps, somewhere, all the ones that I have seen use a pair of throwaway 6 volt lantern batteries, the ones with springs on top.
These have the merits of simplicity and being fairly cheap if purchased in bulk, they are much less used domestically than in years gone by and therefore seldom pilfered.
Rechargeable batteries would not be suited in my view.
Battery life is weeks or months, it varies a great deal according to the quality of the batteries used.
A good quality zinc carbon 6 volt lantern battery should contain 4 cells each "F" size and should have a capacity of at least 10AH. Inferior batteries contain 4 D cells plus a spacer and therefore have a lower capacity.
Alkaline batteries may have a greater capacity, but not always, and may not be worth the considerably higher price.
Air/alkaline batteries exist and have much greater capacities but find little favour due to the higher price, and the limited life once the seal is broken.
EDIT TO ADD was not aware of the preceding post whilst I was typing. The new tail lamp featured DOES make use of air/alkaline batteries of great capacity and hence a long life of well over a year continually.
A cynic like me would suspect that when the expensive air/alkaline batteries are dead that they will be replaced with cheapo zinc carbon ones as used at present.