"Escort" allows for plausible deniability, in case the authorities come nosing around. Yet as it gradually becomes the standard term, the advantage of deniability fades. There is also an economic factor, as escorts tend to cost more.
To some extent "model" has followed a similar trajectory, but the older usage may still be assumed in most cases. "Masseuse" and "masseur" apply to special circumstances.
It is usual to label such terms (understood in their sexual sense) as euphemisms. Yet this broad-gauged explanation fails to capture the semantic shift involved.
A distant parallel is the replacement of the word "lover" (standard in US gay circles twenty-five years ago) with "partner." This has several advantages, suggesting permanence and the recognition that most human relationships are not based exclusively on sex.
I notice that many young heterosexual couples now prefer "partner," even married ones. The advantage seems to be that the new term connotes equality. Seemingly tainted with patriarchal differentiation, "husband" and "wife" still convey a sense of assymmetry.