I'm not quite sure it is what you are looking for but let's give it a try. I run your code, plotting each step so I could analyze graphically the different outputs:

    library(tidyverse)
    library(amt)
    library(sf)
    library(spex)
    
    #' load in the protected areas
    mycrs <- "+proj=aea +lat_1=20 +lat_2=-23 +lat_0=0 +lon_0=25 +x_0=0 +y_0=0 +datum=WGS84 +units=m +no_defs "
    
    merged_Africa = read_sf("pathToFile/WDPA_Jun2020_SWZ-shapefile-polygons.shp")
    st_crs(merged_Africa) <- 4326
    merged_Africa_tranform <- st_transform(merged_Africa, mycrs)
    st_crs(merged_Africa_tranform)

Plotting it I get:

[![enter image description here][1]][1]

The next steps on your code will throw the next result: 

    #' location data
    x_ <- c(707692, 707589, 707998, 708407, 708916, 709415)
    y_ <- c(-3030991,-3031423,-3031640,-3031750,-3032508,-3037158)
    mydata <- data.frame(x_, y_)
    
    # transform to trk
    trk <-
      mk_track(mydata,
               .x = x_,
               .y = y_,
               crs = CRS("+proj=aea +lat_1=20 +lat_2=-23 +lat_0=0 +lon_0=25 +x_0=0 +y_0=0 +ellps=WGS84 +datum=WGS84 +units=m +no_defs"))
    
    #' calculate home range area
    kde_shape_1 <- trk %>%
      hr_kde(., levels = c(0.95))
    hr_area(kde_shape_1) / 1e6 # 52.00846

[![enter image description here][2]][2]

Which I believe is the home area you are defining. Then you do the following:

    psf <- qm_rasterToPolygons(kde_shape_1$ud, na.rm = TRUE)

If we plot it, we get:

[![enter image description here][3]][3]

Which I'm guessing is not what you want and may be the origin of your problems. 
If, instead, we use the function `hr_isopleths` as follows:

    test <- hr_isopleths(kde_shape_1)

And we plot it, we obtain the same case as in the second figure but with the area defined as a polygon we can work with. Calculating the intersection again as:

    intersection <-
      st_intersection(test$geometry, merged_Africa_tranform$geometry)
    sum(st_area(intersection)) / 1e6

We now obtain and area of `34.68964 [m^2]`

Hope it helps 

PS: I'm totally out of my element here, so if it is not what you were looking for, just let me know and I'll delete the answer so others are not dissuaded to participate
  

  [1]: https://i.sstatic.net/wKZcp.png
  [2]: https://i.sstatic.net/cXXDP.png
  [3]: https://i.sstatic.net/5rP0X.png