Finals = Final preparations (plural) to land
I can see the logic of that. But, is it the preparedness to land, from a checklist point of view, or the position in the circuit, which is being reported by the use of the term "finals"?
The 'with you' that is all but endemic in the US is annoying. So superfluous. They're not 'with' them in the sense of being in the same room, and that they're 'with' them on the frequency is self evident by their transmission.
I agree that this may not be a formally accepted phraseology, but yes, it sure is common here. I would expect that it is a contraction of: "[I am now communicating] with you" (I was just communicating with someone else, and was handed over to you, so here I am.). This would only be evident to the new controller
because of the transmission.
I have not flown enough in Europe to have formed an opinion, but in some airspace in North America, it's so busy, you're lucky to squeeze in "XXX Tower NNZ with you". That, as an initial contact, seems to be enough that the controller will call you back when they have time. In my opinion, acknowledging the entry of an aircraft into the airspace might not be as urgent as giving a landing clearance or such, and therefore all the supplementary details can wait until the controller is ready to receive them.
For other airspace I enter, I know that the controller is so bored, I entertain myself by giving very detailed initial contacts with all the appropriate information provided. I know he was waiting pen in hand for someone to call in!
Just my take on it.....