Inextricably linked - inseparable. Previously the term was used as a synonym for the more accurate term 'conjoined twins'.
The term 'Siamese twins' derives from Chang and Eng Bunker (1811–1874) who were the first pair of conjoined twins to become internationally known. The brothers were from Siam, hence the name. In their own country, as they had Chinese parents, they were known as The Chinese Twins.
In 1829, the British entrepreneur Robert Hunter arranged a world tour for the twins, essentially exhibiting them as circus freaks. Posters billing their performances at the time called them the 'United Brothers'. The term 'Siamese twins' began to be used about the same time. Later in the 19th century, 'Siamese twins' began to be used as a general term for conjoined twins.
They used their considerable savings to settle in Carolina in 1839 and fully embraced the southern plantation owner's lifestyle of the day, marrying a pair of local sisters and purchasing slaves to work their farm. In the 1860s, needing money to support their total of 22 children, they joined P.T. Barnum's circus.
The figurative use of the term 'Siamese twins' dates from the 1960s. The earliest printed record of it that I can find is from the Californian newspaper the Pasadena Star-News, March 1963:
"The two organizations [Caltech and the Pasadena Chamber of Commerce] were so closely knit ... they were practically Siamese twins."
See other phrases that were coined in the USA.
See also: Joined at the hip.