This issue goes back a long way... in fact there's a dialog which warns you about this very problem whenever you try to export to SVG (the problem is in Qt)
However it IS possible to work around this. It won't work like an overview, though, so it's not suitable for 'atlas' functionality, I'm afraid. It is tricky and involved. You will probably end up swearing. But it can be done.. here's an example in Inkscape, exported from qgis :)
You can do this by adding two clipped maps, one main one, one inset, and setting the extents manually in composer.
I've only ever tried this with WGS84. Your mileage may vary depending on projection being used. I suspect UTM may be okay but anything with curved meridians won't work.
Manually setting extents and zoom levels
You need to work out the extent you want for the inset map, and meticulously clip the polygons to that extent. This needs a bit of planning, like knowing the zoom level, and it's easier with some projections than others.
- Zoom in to the area you want as the extent of the inset map.
- Make a note of the zoom level
- Use the processing algorithm Clip Vectors by Extent. Choose the canvas extent.
- You should now see something like this.
There, the brown represents the larger area (the inset/context map). The blue represents the zoomed-in area.
- Use metadata to get min/max extent of clipped layer e.g. xMin,yMin -29.4264,-20.1055 : xMax,yMax 18.7455,26.9695
- turn off all layers except the clipped one.
- add a new map. shape doesn't matter as the next two steps take care of this.
- copy and paste the scale into the Composer item settings
- copy and paste these 4 values (xmin, ymin, xmax, ymax) manually into the extents on the Composer Item settings. Don't use set to map canvas extent. It's tempting, but it may break things.
- Add a frame (if needed), grid etc.
- Lock layers (optionally, Lock style for layers)
repeat this process for the map zoomed in (making sure you turn off the clipped layer this time). You can use Layout > Send to back to slide it under the inset map.
Finally, export as SVG from composer. You might find it helps to use the clip to content option here.
Not quite sure how to get the 'highlight' to appear inside the inset map, but I hope this gets you closer to the output you're after.