Van Nuys Tower near Los Angeles is another class D that gives VFRs beacon codes. Some facilities just work that way, even though it's technically not required to have a transponder to go into most Class D's. At a VFR tower I once worked at, I gave out codes to inbounds one evening because I couldn't keep track of the non-stop arriving VFRs (over 70 ops to a single runway that hour).
With the flight following thing, if you specify that you want flight following "all the way to ABC airport", you'll probably have a higher success rate that just saying that you're going to ABC airport. Some facility SOPs require controllers to provide radar services only until a specific distance from the airport (usually 20 miles for class C airspace, for example), unless other services (like flight following) are specifically requested.
Reason for editing: Discussed scenario with another controller who pointed out that our own facility SOP tells us that radar services should be terminated 20 miles from the airport except for when requested by the pilot.