I don't have Delphi, so I don't know what Dephi's zlib unit contains.
Looking at the source you sent me, the solution whould be to use the zstream unit, included with fpc FCL.
In fpc the zlib unit is simply a wrapper for the functionality in z compression library, which is comonly available on unix system as libz.so.
Fpc also has a paszlib unit (source in packages\base\paszlib) which is a pascal implementation of the z-compression functions, so this unit does not depend on external libraries. The zstream unit (containing among others TCompressionStream) uses this unit.