The Lego Group, Denmark

This image [above] is dated 2017.

[The company's] current and formal name is Lego System A/S (Aktieselskab, English: stock company or limited company) using the trade name The Lego Group.
Source: https://www.lego.com/en-us/lego-history

It has been common practice to display the corporate logo and its respective flag on the model kits sold by the company, which of course includes not only the current logo, but also its past versions.
Esteban Rivera, 06 December 2020

The current company's logo (http://danishlogo.dk/wp-content/uploads/2017/12/LEGO.png, source: http://danishlogo.dk/?portfolio=lego&pnt=366) was designed in 1973 by Danish Graphic Designer Niels Hartmann and Swedish Graphic Designer Rolf Lagersson. "A single new LEGO logo replaces the various logos that have been used until now. The new LEGO logo unifies all the company is products under one banner."
Sources: https://www.lego.com/en-us/aboutus/lego-group/the-lego-group-history/
and https://www.lego.com/en-us/aboutus/lego-group/the-lego-brand/
Esteban Rivera, 18 July 2020

I don't have any sources at hand, either, but that would surely be "ludo"? Concerning "lego" the Wiktionary says:
- I choose, select, appoint.
- I collect, gather, bring together.
- I take, steal.
- I traverse, pass through.
- I read (aloud), recite.

Regrading the flag, the letters are a special commissioned typeface, very rounded bold serifless capitals, indeed white but separated from the red background by a double thick fimbriation of black and yellow, the black filling the eyes of the lettereforms in such way that there's no isolated yellow or red areas.
This flag design has been available as one of the building blocks itself, to be used in lego buildings, since at least 1973 and as a decal to be glued on blank flags.
I generated a 216 px high rendering of the logo from the Wikimedia image and used, recanvased for flag image.
António Martins-Tuválkin, 21 May 2012

There's a picture showing the flag of LEGO, a popular line of construction toys, consisting of colorful interlocking plastic bricks and an accompanying array of gears, minifigures and various other parts. The source mentions that the highest Lego tower was built in Seoul, Korea, in 2012.
Source: http://noticias.latam.msn.com/co/fotos.aspx?cp-documentid=33764773

The Lego (trademarked in capitals as LEGO) Group began in the workshop of Ole Kirk Christiansen, a carpenter from Billund, Denmark, who began making wooden toys in 1932. In 1934, his company came to be called "LEGO", from the Danish phrase leg godt, which means "play-well"(Coincidentally, "Lego" also means "I assemble" in Latin.). Its known as The Lego Group since 1932.
Source: Wikipedia

The flag is a red horizontal background with the name in white capital letters, showing this logo.
For additional information go to: LEGO (official website)
Esteban Rivera, 14 May 2012

Others say (MERIAN, don't yet know the year), that LEGO is Latin, meaning: "I play".
Klaus-Michael Schneider, 14 May 2012

Based on the company's own information, and connected time line pages:
Ole Kirk Kristiansen founded The Lego Group in 1932 and as it was passed from father to son, it is currently owned by Kjell Kirk Kristiansen, grandson of the founder. The company started out as a workshop producing wooden household appliances and toys, but the share of the toys quickly grew. During the nineteen forties the company went plastic, and introduced the "Automatic Binding Brick". In 1958, the current binding system for the bricks, by then simply called "LEGO bricks", was patented, and in 1963 the current material was introduced; the two improvements together creating very tightly binding bricks. The system of the bricks was extended to included similar elements, to include moving parts, and to include models built from LEGO parts, and eventually came to include quite a few other types of toys etc., in a LEGO style.
Already in 1934, though still producing other products as well, the company and its products were named "LEGO", formed after the Danish "Leg Godt" (play well). In 1954 the word "LEGO", in capitals, was registered as a trademark. The year 1973 saw the the company introducing the single LEGO logo, for all its now rather diverse products, a logo which was in use unaltered for 25 years. In 1998, a new graphic for the logo was introduced. (Both variants might appear on flags.)
At some point after the introduction of the building bricks, it was realised one could read "lego" as Latin for "I assemble". (Somehow, the Latin reference in some cases got mixed up with "ludo": "I play".)
Peter Hans van den Muijzenberg, 14 May 2012