I would say yes, if the cabinets look like regular furniture. Your designer and you will have to get that sorted out. When I put "Can I extend kitchen cabinets into the dining room? in photos", there are some ideas for you.

Thanks. I’ll search some more. I must not have been using the correct terms.

What if you extend the cabinets next to stove and make a breakfast bar?

When you are dealing with a small kitchen with a nearby dining area, I would skip a breakfast bar. Sitting on a stool with feet tucked wherever you can manage to tuck them is less comfortable than sitting on a chair with feet on the floor. A kitchen needs prep space, storage space (for food and everything else), space to use countertop appliances, and pantry storage. You need space next to the sink for both prep and cleanup. When I empty the dishwasher, I sort and stack all dishware on the counter and want to have storage very near the dishwasher...cookware in lowers, dishes and glassware in uppers.
Yes, go ahead and extend your kitchen storage into the dining space: many people no longer use or want a formal dining room. I would keep the countertop appliances in the kitchen zone and run closed-door cabinetry along the sink wall. You don’t want to block that far corner window, so the last cabinet may need to end before reaching the corner.
Can you post photos of the actual rooms?

Yes, but move some appliance to the extended area so it has a function. Also move window if possible.

An island should be in consideration ...more cabinets on the walls will most likely make the space look and feel even smaller

Leaving the open space unuseable

I’m not sure where an island would go. I can’t picture it. The pics I have the rooms are still full. Sorry.

wherever the person stood ...in taking the first picture ...is the general area for the island...it is very hard to tell where the second picture is located from the floor plan...

I seriously question if the table and chairs will even fit as proposed..in a space only 8’9” wide?...

its 9’ not 8’9 I don’t know why the program keeps changing it. But anyway, the dining room has always been that size and had a table and chairs in it.

the second photo is the dining room taken from the opposite side, facing the wall i wanted to add cabinets to.

If that window is in play at all, I'd replace it with a counter height one, or remove it altogether, and take the cabinets all the way through that corner in an L. I'd make the entire bottom rectangle kitchen. The dining area is tight, but you say you already have a table and chairs there and it works for you. What's the rest of the home's layout look like?