Connotations arbitrary and invisible are connotations that don’t really matter anyway, but let’s get some idea of the issue reading some excerpts from the Wikipedia.
The search for a symbol began in 1950 when a committee was set up in order to look into the question of a European flag. There were numerous proposals but a clear theme for stars and circles emerged.
Richard von Coudenhove-Kalergi proposed that they adopt the flag of his International Paneuropean Union, which was a blue field, with a red cross inside an orange circle at the centre, which he had himself recently adopted for the European Parliamentary Union. [See Letter to the secretary general of the Council of Europe from Richard Coudenhove-Kalergi, Council of Europe.]
Due to the cross symbolism, this was rejected by Turkey (a member of the Council of Europe since 1949). Kalergi then suggested adding a crescent to the cross design, to overcome the Muslim objections. [See Letter from Richard Coudenhove-Kalergi regarding a Muslim modification to the Pan-Europa flag design, Council of Europe.]
The Consultative Assembly narrowed their choice to two designs. One was by Salvador de Madariaga, the founder of the College of Europe, who suggested a constellation of stars on a blue background (positioned according to capital cities, with a large star for Strasbourg, the seat of the Council). He had circulated his flag round many European capitals and the concept had found favour.
The second was a variant by Arsène Heitz, who worked for the Council’s postal service and had submitted dozens of designs; the design of his that was accepted by the Assembly was similar to Salvador de Madariaga’s, but rather than a constellation, the stars were arranged in a circle. In 1987, Heitz claimed that his inspiration had been the crown of twelve stars of the Woman of the Apocalypse, often found in Marian iconography. [See “Real politics, at last”. The Economist. 2004-10-28.] …
The French satirical magazine Le Canard enchaîné reacted to Heitz’s statement with an article entitled L’Europe violée par la Sainte Vierge (“Europe Raped by the Blessed Virgin”) in the 20 December 1989 edition. Heitz also made a connection to the date of the flag’s adoption, 8 December 1955, coinciding with the Catholic Feast of the Immaculate Conception of the Blessed Virgin Mary.
Paul M. G. Lévy, then Director of Information at the Council of Europe responsible for designing the flag, in a 1989 statement maintained that he had not been aware of any religious connotations.
In an interview given 26 February 1998, Lévy denied not only awareness of the “Marian” connection, but also denied that the final design of a circle of twelve stars was Heitz’s. To the question “Who really designed the flag?” Lévy replied:
“I did, and I calculated the proportions to be used for the geometric design. Arsène Heitz, who was an employee in the mail service, put in all sorts of proposals, including the 15-star design. But he submitted too many designs. He wanted to do the European currencies with 15 stars in the corner. He wanted to do national flags incorporating the Council of Europe flag.” [See Carlo Curti Gialdino, I Simboli dell’Unione europea, 2005. pp. 80–85.]