How to extend the boundary of polygon until a line?

I have two objects:

eez:https://github.com/yuliaUU/test/blob/main/eez.RDS

bordering_polygons: https://github.com/yuliaUU/test/blob/main/madg_bordering_polygons.RData

So what I want to do is to try to extend the outer boundary of each bordering_polygons untill eez line, and so that boundaries will not overlap

As suggested , i tried teh following:

v <- vect(bordering_polygons)
r <- rast(eez, res=.1)
# get the distance to each polygon, and then find the nearest polygon
rr <- lapply(1:nrow(v), \(i) distance(r, v[i])) |> rast() |> which.min()

z <- erase(as.polygons(rr), v)
values(z) <- values(v)[unlist(z[[1]]),  ]
x <- rbind(v, z)
a <- aggregate(x, names(x))
plot(v)
plot(a, "ADM1_EN", type="classes")
lines(v, lwd=3, col="gray")
lines(eez, col="red", lwd=3)

my only issue that the created areas are not combined with existing polygons

The tricky bit is to extend polygons. Here is an approach that uses a (discrete) raster.

Example data (please do not ask questions based on files that need to be downloaded)

library(terra)
# your island 
v <- vect(system.file("ex/lux.shp", package="terra"))
# use planar crs to speed things up
vcrs <- crs(v)
crs(v) <- "local"
# example EEZ
e <- as.polygons(ext(v) + .05, crs=crs(v)) 

Define a raster (higher resolution is better, but slower)

r <- rast(e, res=.01)
# get the distance to each polygon, and then find the nearest polygon
rr <- lapply(1:nrow(v), \(i) distance(r, v[i])) |> rast() |> which.min()

Create polygons, erase what we already have, set the attributes and combine with the admin areas of the island.

z <- erase(as.polygons(rr), v)
values(z) <- values(v)[unlist(z[[1]]),  ]
x <- rbind(v, z)
a <- aggregate(x, names(x))
crs(a) <- vcrs

Illustration of the result

plot(a, "ID_2", type="classes")
lines(v, lwd=3, col="gray")
lines(e, col="red", lwd=3)

For your data you need some minor adjustments (because the island has a hole and the outside boundary is not rectangular.

eez <- vect(eez)
mdg <- vect(bordering_polygons)
mcrs <- crs(mdg)
crs(eez) <- crs(mdg) <- "local"

r <- rast(eez, res=.1)
rr <- lapply(1:nrow(mdg), \(i) distance(r, mdg[i])) |> rast() |> which.min()

fmdg <- aggregate(mdg) |> fillHoles()
z <- erase(as.polygons(rr), fmdg)
values(eez) <- NULL
z <- intersect(z, eez)
values(z) <- values(mdg)[unlist(z[[1]]), ,drop=FALSE]
x <- rbind(mdg, z)
x <- aggregate(x, names(x))
crs(x) <- mcrs

plot(x, "ADM1_EN", type="classes")
lines(mdg, lwd=3, col="gray")
lines(eez, col="red", lwd=3)