TLDR: Yes

Firstly, the construction of the Statue of Liberty was primarily organized by private citizens, both American and French. The statue was actually very well advertised because of fundraising efforts for the statue, which was mostly done in France, and the pedestal, which was mostly done in the USA.

The first people to know about the plan were probably mostly abolitionists but as a site was being chosen, local governments along with President Grant all learned about the plan. The project was officially announced in 1875 and public fundraising began.

President Hayes earmarked the land for the construction of the pedestal while construction began in both France and the USA.

In France, there were multiple political motivations for donors, the statue was meant to solidify relations between the USA and France, it was also a reminder of French assistance during the American Revolution, it was a symbol of French strength, and it was a symbol of their ability to bring light and liberty to the rest of the world (at the time a common justification for imperialism was that it brought an end to local injustices like slavery and tribalism by introducing western law)

In the USA, fundraising had it’s own interesting quirks. Firstly, the construction of the Washington Monument was repeatedly stalled so it may have seemed unusual to some people to start work on another project, especially during an economic crisis. Additionally, the US was still fresh out of the Civil War which meant that there were plenty of Americans who were excited about the opportunity to have a monument to liberty in New York.

The construction took years and throughout that time, there was a nearly continuous campaign to raise public awareness of its construction. Additionally, American and French political leaders were involved throughout the process which involved multiple public ceremonies.

To conclude, while the Statue of Liberty was a gift, a better term might be “joint project”.

Fun fact: the Statue of Liberty was never a shiny copper color. The patina had already started to form on the parts before the final assembly began and it was mostly green by the time it was dedicated. The color change was expected and actually protects the metal from further damage which is part of the reason that it’s never cleaned off,