If it helps to frame your thinking, put away the "I'm being a nice neighbor" hat for a minute and ask yourself how you'd feel about buying a place where the neighbors AC was run though your space. It could knock 10s of kilobucks off your resale, I'm thinking.

This question has two parts, and both are missing critical details. For a physical answer you'd have to provide a sketch and perhaps some pictures. Also highlight any (shared) utility entry points. For a legal answer you'd have to specify your jurisdiction, and supply pertinent sections of your deed / land registry. In any case, as interesting as it is, it cannot be answered here as asked.

Why not run the refrigerant lines on the outside of the building? Have you or your neighbour asked the city? Sometimes you can get permission in special cases by asking the city directly to waive the bylaw preventing installation when you have no other options. The driveway obviously isn't an option, but the "too close to the property boundary" is something they might let them get away with if you get permission in advance. The space between the buildings seems otherwise the ideal spot so I wouldn't give up on that without at least asking the city first.

As an example, I have a sun porch on the side of my property that was added by the previous owners and which is also "too close to the property line" - they applied for special permission to build it anyway (I have the documents) and it was granted with no objections. Usually building close to the line is disallowed in general, but for special cases it can often be waived. This should be your first objective since it is by far the best solution. You may need permission from the neighbours on their other side, but it's way better "too close" to theirs than running straight through yours.

If this is a two-story unit then your neighbor could hang a min-split condenser unit high enough over the driveway. It's possible to have multiple rooms run off of a single mini-split condenser. Would the "other codes" prohibit such an installation?

@DaveD Yeah, not a bad idea. They can even go central A/C with a side-mount unit (like the Daikin FIT, for example). Could bolt right onto the house between the units in the "no space" zone.

Can your neighbor mount the condenser (outside) unit on the roof?

And if everything OP said is true, the local jurisdiction is basically telling OP's neighbor that she can't have an AC unit on the ground outside her house, but OP can. That just doesn't sound right to me. Maybe OP's neighbor just hasn't tried hard enough, like seeking a variance from the planning/zoning people.

@J... Odds are the property is built up to the easment. It's not "forbidden" to build on an easement, you just have to accept that at any time the city can destroy what you built there, and bill you for destroying it. From an asthetic point of view, it looks like the backyard is driveway, which maybe can't be narrowed, the front is "front yard" which is a bad idea even if legal, and the side is just very narrow. Get them to look at side venting units or roof units. I had a roof unit, it never gave me a day's worth of trouble (beyond wearing out).

it is quite possible that you do not own the attic space ... maybe the lines can run through there

@P2000 I've added an image as requested. The jurisdiction is Kirkland, King County, WA.

I know people who have been allowed to put AC units on a public sidewalk because of a "hardship" plea.

@Vivek - Has your neighbor applied for a variance? That should be the first approach. Like others have said, almost all jurisdictions have a process whereby bulk requirements (like building setbacks) can be waived. It happens all the time in my neck of the woods.

@EdwinBuck OP has made no mention of an easement. I think that's speculation.

@SteveSh They didn't think about that. I passed it along to them, along with a bunch of other helpful ideas that came from these posts. Ty

To be clear, are the red rectangles which indicate your neighbor and you supposed to indicate buildings? And the property they are on is shared? Do you outright own the townhouse you are in, from foundation to roof? Or is part/all of it shared/leased/rented or outright owned by someone else? There are probably regulations about shared access in your municipality since you are in a townhouse and so close to a road (in fact I'm sure there are since you mention regs that say you can't put A/C that close to the street). These regs surely mandate how these HVAC lines, etc. need to be run.

@TylerH It's a duplex - OP owns the right side, their neighbour the left. They share a roof and a dividing wall but each will own the land only on (and around) their side. The private driveway leads to four such duplexes (two south side, two north side). The four units also likely have some sort of shared ownership of the driveway and the usual condo-style association to manage those shared resources.

@J... You're guessing at that unless OP has clarified in a now-deleted comment, and that information somehow didn't make it into the question.

@J... Actually, I see now for some reason OP replaced the photo with a diagram that is actually less accurate and detailed...

@TylerH Not guessing.. The private has a condo corp but the unit comes with 0.463 acres of land. And yeah, the google map photo I had added to the question but identified the wrong building in that photo - thought it was the SE building of the four but it's actually the SW building.

@VivekMaharajh Why do you say there is "no space here" in the area circled in blue? Looks like there is plenty of space to me.

@TylerH That's where OP's A/C is. The neighbour is on the other side. They'd have to put it at the street here - right in the garden. The area in blue is not the correct spot - that was my misidentification of the unit.

@TylerH You can if you like. The issue with room is strictly city regulations - obviously there's physical space there. The problem was that OP's neighbour did not consider anything other than a top-exhaust unit and that puts them too close to the sidewalk (by the rules). As everyone else here has said, they can likely get a variance either way.

@TylerH It looks like you put the wrong image back - OP's unit is the southwest one, not the southeast one as indicated in that image.

@J... Yes the edit was made before our interaction here.

I'd be curious what other duplexes in the community have done. Or is this the first one anywhere in the neighborhood with the issue?

@TylerH yes, the red rectangles indicate buildings. Yes the property is shared. I own the interior of the townhouse. I'm not sure if I own things like the roof.

@TylerH I didn't say there's "no space here". That is actually incorrect. That is where my AC is currently, and there's abundant space there for another AC. I removed the image because it was incorrect, but it looks like someone added back the image with the inaccuracies...

I'd be happy to update the drawing to have whatever useful information you think is in the image that isn't in the drawing so that we can just have one image without any inaccuracies...