A call girl is a woman who arranges paid sex by telephone and visits a client indoors, rather than finding clients in the street. The phrase dates from the middle of the twentieth century, when the telephone was the thing that made the distinction possible.
What the word was distinguishing
It marked a split in the trade. Street work meant being visible, being arrested and taking whoever turned up. Working by telephone meant screening, privacy, higher rates and much less exposure to the police. The word carried that difference, and with it a suggestion of being higher up the trade than the alternative.
Why it sounds dated now
Because the telephone stopped being the distinguishing feature. Almost everything is arranged online, screening happens by message, and nobody is defined by owning a phone. The word survives mostly in films, headlines and nostalgia.
What replaced it
Escort, in almost all ordinary use. It covers the same arrangement, carries less of the period flavour, and allows for bookings that are social rather than sexual. What an escort is covers the modern version, and courtesan, mistress and madam covers the older words that sit alongside it.
The law does not care about the word
In England and Wales the legal position depends on what is done and where, not on what it is called. Paying another adult for sex in private is not an offence. Soliciting in a street or public place is. So is running premises where more than one person works, and so is controlling someone for gain. The UK law guide covers it, and escort or prostitute covers how the words are used today.