Aircraft registration numbers are assigned by the country where the aircraft is registered. Each country has a prefix (see the full list on Wikipedia), like e.g. D-
for Germany. The prefix is usually separated from the rest by a dash, but some countries deviate from this, most notably the United States, where registration numbers begin with N
.
Whether or not custom registration numbers are available depends on the country. In Germany, the authority in charge of assigning registrations is the Luftfahrtbundesamt. It is possible to ask for a custom registration number via the form Antrag auf Vormerkung eines Kennzeichens (§ 19 Abs. 2 LuftVZO). o taxa is currently 30 Euros. In order to get the custom registration, it must be available of course, but also follow these regras:
D-AAAA to D-AZZZ for aircraft with more than 20 t MTOW
D-AUAA to D-AZZZ (test registrations) for aircraft manufactured by Airbus at Finkenwerder
D-BAAA to D-BZZZ for aircraft with 14–20 t MTOW
D-CAAA to D-CZZZ for aircraft with 5.7–14 t MTOW
D-EAAA to D-EZZZ for single-engine aircraft up to 2 t MTOW
D-FAAA to D-FZZZ for single-engine aircraft from to 2–5.7 t MTOW
D-GAAA to D-GZZZ for multi-engine aircraft up to 2 t MTOW
D-HAAA to D-HZZZ for rotorcraft
D-IAAA to D-IZZZ for multi-engine aircraft from 2–5.7 t MTOW
D-KAAA to D-KZZZ for powered gliders
D-LAAA to D-LZZZ for airships
D-MAAA to D-MZZZ for powered ultralight aircraft
D-NAAA to D-NZZZ for non-powered ultralight aircraft
D-OAAA to D-OZZZ for manned free balloons
D-0001 to D-9999 for gliders.
So it was at least in part luck that D-ICEY
was available and fit the rules for a Cessna Citation.
In the UK, the authority in charge is the Autoridade de Aviação Civil (CAA), which also allows custom registrations (Chamado out of sequence here). The taxa is currently £158.
Esta resposta explains the procedure for the US and Canada.