With thanks to Paul Norton - UK and Lýður Pálsson (museum curator) - IS for their support.

Iceland was part of Denmark, but is an independant republic since 1944. That is why this island still has strong bonds with Denmark, what is a.o. shown by the typical edition called Matador. However, the question is where its precursor Milljóner was invented, in Denmark or Iceland or Norway perhaps? 
It was not until 1997 Parker Brothers/Hasbro also introduced their version "ref.14535" with of course the streets of the island's capital Reykjavík. 

< Advertisement from the daily Alþýðublaðið of  20-Des-1939.
   Oldest mention of Miljóner (Matador), by the "New Toyfactory" in Reykjavik.

Thómas V.Albertsson, a Folkloristic from Iceland, recently informed me about the following poem, used to promote the Matador game in the days before Christmas 1939:

The poem goes like this: 

The spaces of the Hasbro edition from 1997 are from BYRJUN onwards:

Edition: Milljóner             
Publisher: Nýa Leikfangagerðin, sími 3749 (The new toy factory, tel. 3749)
                  Skólavörðustíg 18 (in Reykjavík) - ±1940    
Dimensions of the game board: 45 x 45 cm
The game:
This edition is property of and shown under nr.B.Arn. 1971 in the Húsíð - Folk Museum  "The House of Merchants" in Eyrarbakki, Iceland.
Lýður Pálsson tells about it: 

"This set was used in the farm Galtastadir in the Gaulverjabær district. It is most probably issued around 1940 and it cannot be younger than from 1944,  because it is known it was played by children who were born 1925 - 1935. "

From Lagt af Stað (Begin) onwards the other 39 numbered spaces are resp.:   

Fjölnisvegur - Freistið Gæfunnar (Surprise) Garðarsstræti - Fangelsi  ( - Borgið 10% Tekjuskatt eða 200 Kr - Freistið Gæfunnar - Blái borðinn veghúsastíg - Vesturgata - Fríkirkjuvegur - Nýja Leikfangagerðin - Hafnarstræti - Túngata - Kexverksmiðjan Frön - Sîlvellir -  Liverpool - Sjafnargata - Freistið Gæfunnar - Bankastræti - Tjarnargata - Sundhöll (= Swimming pool) - Pósphússtræti - Tryggvagata - S.Í.S. - Freistið Gæfunnar - Laugavegur - Ölg. Egill Skalla Grímss. - Asturstræti - Skólavörðustígur - Fatabúðin - Bankastræti - Freistið Gæfunnar - Aðalstræti - Borgið Aukaútsvar (Pay extra tax) - Eimskip - Austurvöllur - Freistið Gæfunnar and Lækjartorg. 

The spaces are in a circle over the solid game board, the background of which is green. There is an illustration outside the circle in each corner: a red car next to Begin - a puppet built of wooden rings and the word Uppeldisleikföng (= educational play toys), on top of the manufacturer's address - the exterior of the swimmingpool of number 21 and a building next to number 31 (= Fatabúðin).
There are only 7 colored groups of properties, and there are two Bankastræti (on nr. 19 and 32). This is the only set so far I've met in which there are 2 of the same street. I have no explanation for that.
There is no castle for the most expensive street, like in the Norwegian edition.

Features of this issue are:

Edition: Matador
Publisher: Inland manufacturer Kassagerðin  -  ±1970
Dimensions of the box: 23 x 44.7 cm 
The game:
Despite the great resemblance with the Danish game a great difference is obvious. On the red lid is only the word Matador, without the illustration of a hand towing 4 rods connected to a boat, hotel, works and an apartment building (see Denmark). 
On the soft gameboard with green playside are the 40 numbered spaces in a circle. The fold goes through the vertical word Matador in the centre of the board. The streets are from the capital Reykjavík. 
The spaces are from Birjid hér onwards: 

Reynimelur - Freistid Gæfunnar - Vídemelur - Greidid 10% Eignaskatt eda 2000 - Flugfél Ísl. - Sólvallagata - Freistid Gæfunnar - Hofsvallag - Hávallagata - Fangelsi - Sjafnargata - Hitaveitan - Fjölnisvegur - Laufásvegur - Samvinnutr - Barónstígur - Freistid Gæfunnar - Skólavördus - Hringbraut - Pér Vinnid 1000 I happor - Gardastr. - Freistid Gæfunnar - Túngata - Vesturgata - Loftleidir - Hverfisgata - Laugavegur - Rafmagnsv - Lækjargata - Í fangelsi - Adalstræti - Hafnarstr. - Freistid Gæfunnar - Pósthússtr. - Sjóvá - Freistid Gæfunnar - Bankastr. - Greidid 1000 aukaútsvar and Austurstr.

Comparison of this Matador edition with the earlier issued Milljóner shows:

The only 25 yellow "Chance" cards still stick together on one sheet and hence they  have to be detached.
All banknotes have the same pattern and are in 6 denominations: 10 - 50 - 100 - 500 - 1000 and 5000.
The tokens are 6 small plastic cars. The green, plastic houses and red hotels are hollow.
Both dice are black with gold pips.

Some remarkable differences between this Icelandic game and the Danish edition are: 

This Matador edition is also shown under nr.B.Arn. 2165 in the earlier mentioned Húsíð - Folk Museum

The set in the red box was bought in August 1975 for IKr 645.- in the bookshop BÓKABÚÐ MM in Reykjavik.

Edition: Standard, ref. 14535.182   
Publisher: Parker Brothers/Hasbro Nordic Denmark -1997
Dimensions of the box: 26.8 x 40.3 x 5.3 cm
of the folded game board: 25.2 x 25.2 cm
The game: 
This is the first Hasbro issue in Iceland. Only the 3 properties of the yellow group can be found again in the older games described here above.

In this so well-known "standard" box is of course the red plastic "banker's tray" with holes for the 7 banknotes and property deeds still present with the banker.
The game board is folded in four and fits, when stored, precisely on the banker's tray. The back of the board is as usual red, the playside is blue green. This edition is made in Ireland and therefore it shows the picture of the "laughing family" at the bottom of the box.
On the game board the Áhætta cards with nice red back are as usual near the BYRJUN corner while the Samfélagssjóður cards with their nice blue back are near the Free Parking corner.
The banknotes are of the one side printed new model, i.e. with a black Monopoly ribbon and Uncle Pennybags (since 2000 it is officially Mr. Monopoly) in the upper part of the circle and, almost invisible, the remark "1996 Tonka Corporation" at the bottom. The currency is Kr. and the denominations are resp.: 100 - 500 - 1.000 - 2.000 - 5.000 -10.000 and 50.000.
The 10 tokens are of light metal and are the well-known: hat - shoe - car - boat - dog - wheelbarrow - rider-on-a-horse - thimble - iron and canon. The green houses and red hotels are of not transparant plastic and have all an overhanging roof with an excentric chimney.
Both dice are white with black pips.

The price of this set was IKr. 3990.- (± € 45.-) June 2001.

It is strange that in this edition some new properties are introduced. (The underlined fields are unchanged.)

The properties are now from BYRJUN onwards:

Bárugata - Samfélagssjóður - Brávallagata - Tekjuskattur borgaðu - BSÍ Umferðar-Midstöðin - Kaldalind - Áhætta (Kans) - Jöklalind - Haukalind - Fangelsi/Bara í Heimsókn - Stórhöfði - Rarik - Höfðabakki - Bídshöfði - Egilsstada-Flugvöllur - Lyngháls - Samfélagssjóður - Fossháls - Krókháls - Frítt stæði - Ármúli - Áhætta - Síðumúli - Suðurlandsbraut - Akureyrar Flugvöllur - Bankastræti - Austurstræti - Orkveita Reykjavíkur - Laugavegur - Farðu í fangelsi - Mávanes - Haukanes - Samfélagssjóður - Blikanes - Leifsstüð - Áhætta - Smáralind - Fjármagnsjuskattur borgaðu and Kringlan.

The banker's tray of this standard edition is red again and has slots for all 7 banknotes and the property deeds still with the banker. 
The board is folded in four and fits, when stored, precisely on the banker's tray.The back of the board is black and that of the playing side is blue green. 
On the board the Áhætta cards with red back and ?, are as usual next to the Start corner and the Samfélagssjóður cards with their blue back and treasury next to the Free Parking corner.The cards (56x87 mm) do not have rounded corners.The texts on both cards are  printed in red.
The property deeds (56x87 mm) do not have rounded corners.
The banknotes (52x100 mm) are printed single-sided. Up in the circle on the banknote is a black ribbon containing "MONOPOLY with Mr.Monopoly". At the bottom of this circle is in lowercase "©2001 Hasbro Inc.". The currency is Kr and the values of the 7 banknotes are resp.: 100 - 500 -1.000 - 2.000 - 5.000 - 10.000 and 50.000 (ref.100 00009 00A0).
The 10 tokens are of light metal and are the well-known: top hat - shoe - car - boat - dog - wheelbarrow - rider-on-a-horse - thimble - iron and canon. 
The houses and hotels are of opaque plastic and all have an overhanging roof as well as an off-center chimney.
Both dice are white and have black pips.

The price of this edition was in october 2010 IKR 5.199 (€ 33,75).