I'm in the US so I use permanent marker to write my lawyers phone number on my arm before protests
And yet as far as I can tell, most middle class Americans seem to refer to "their lawyer". Do you pay a monthly fee? Are they a criminal defence lawyer, or something broader? How often do you talk to them? How do you find them?
It is possible to have an attorney on retainer though, either as a consequence of having hired that attorney in the past or as part of a subscription service.
I've only run into this among the so-called "upper middle class" here (e.g., physicians making $500K+/yr) and even then it's pretty rare.
Funnily enough, Americans do not use the term solicitor; that's reserved for lawyers working for the government!
It is certainly a rare term in American English. I associate it with the probably now-archaic "NO SOLICITORS" signs, which used to be commonly used in an effort to ward off door-to-door salesmen and such. The specialized usage you are referring to is the use in titles of certain important government lawyers (I'm only aware of this in the federal government). The most famous is the Solicitor General, which is an appointed official in the Department of Justice whose job is mainly to argue on behalf of the government before the US Supreme Court.