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But Argentina nor Spain never had any settlers there before. The French came first then finally settled by the British.

Literal Spanish boots on the ground, sure, but they did own them. And the French had given them back to the Spain, who owned them by treaty.

From the wiki ...

Spanish settlement

*In 1493, Pope Alexander VI issued a Papal bull, Inter caetera, dividing the New World between Spain and Portugal. The following year, the Treaty of Tordesillas between those countries agreed that the dividing line between the two should be 370 leagues west of the Cape Verde Islands.[6] The Falklands lie on the western (Spanish) side of this line. *

Spain made claims that the Falkland Islands were held under provisions in the 1713 Treaty of Utrecht which settled the limits of the Spanish Empire in the Americas. However, the treaty only promised to restore the territories in the Americas held prior to the War of the Spanish Succession. The Falkland Islands was not held at the time, and were not mentioned in the treaty. When Spain discovered the British and French colonies on the Islands, a diplomatic row broke out among the claimants. In 1766, Spain and France, who were allies at the time, agreed that France would hand over Port Saint Louis, and Spain would repay the cost of the settlement. France insisted that Spain maintain the colony in Port Louis and thus prevent Britain from claiming the title to the Islands and Spain agreed.[5] Spain and Great Britain enjoyed uneasy relations at the time, and no corresponding agreement was reached.[4]

The Spanish took control of Port Saint Louis and renamed it Puerto Soledad in 1767. On 10 June 1770, a Spanish expedition expelled the British colony at Port Egmont, and Spain took de facto control of the Islands. Spain and Great Britain came close to war over the issue, but instead, concluded a treaty on 22 January 1771, allowing the British to return to Port Egmont with neither side relinquishing sovereignty claims.[7] The British returned in 1771 but withdrew from the islands in 1774, leaving behind a flag and a plaque representing their claim to ownership, and leaving Spain in de facto control.[8]: 25 

From 1774 to 1811, the islands were ruled as part of the Viceroyalty of the River Plate. In that period, 18 governors were appointed to rule the islands. In 1777, Governor Ramon de Carassa was ordered to destroy the remains at Port Egmont. The British plaque was removed and sent to Buenos Aires.[5]: 51 

Spanish troops remained at Port Louis, known then as Port Soledad, until 1811[9] when Governor Pablo Guillen Martinez was called back to Montevideo as the revolutionary forces spread through the continent. He left behind a plaque claiming sovereignty for Spain.[4][10]

Basically Spain owned the islands, found out later that the French and English were land squatting and had moved in on their islands, and had something to say about the matter. The French gave their land back to Spain, the English did not.

There's allot of history and conflict over the CENTURIES there to unpack. Its a nuanced conversation.