In the comments mfvonh correctly points out that the Wikimapia's URL in my question is a user interface URL, not a tile server URL. The tile server link template is documented in a message which appears when one sends incorrect request to the tile server:

From the above it is clear that the StringTemplate
should be as follows:
StringTemplate["http://i<*Mod[#2,4]+Mod[#3,4]*4*>.wikimapia.org/?x=`2`&y=`3`&zoom=`1`"]
The tile server also accepts lng
parameter which specifies the language of the map (addition of &lng=0
selects English) and type
parameter which specifies the type of the map: hybrid
(semi-transparent) or map
(the default for the tile server, doesn't have alpha-channel). The largest supported zoom level is 22, so we should add the "ZoomRange" -> {1, 22}
option. Wikimapia as well as Google and OpenStreetMap uses the
EPSG:3857 - WGS 84/Pseudo-Mercator (Spherical Mercator)
projection which seems to be the default for GeoGraphics
, so we may not specify it explicitly.
Testing:
GeoGraphics[GeoRange -> Entity["City", {"Paris", "IleDeFrance", "France"}],
GeoServer -> {StringTemplate[
"http://i<*Mod[#2,4]+Mod[#3,4]*4*>.wikimapia.org/?x=`2`&y=`3`&zoom=`1`&lng=0"],
"ZoomRange" -> {1, 22}}]

Let us try to use the Wikimapia API for highlighting the basement of the Eiffel Tower:
data = Import[
"http://api.wikimapia.org/?key=example&function=place.getbyid&id=55&format=json&pack=&\
language=en", "JSON"];
GeoGraphics[{EdgeForm[{DotDashed, Blue, Thick}], FaceForm[],
Polygon[("polygon" /. data)[[;; , ;; , 2]]]},
GeoServer -> {StringTemplate[
"http://i<*Mod[#2,4]+Mod[#3,4]*4*>.wikimapia.org/?x=`2`&y=`3`&zoom=`1`&lng=0"],
"ZoomRange" -> {1, 22}}, GeoRangePadding -> Scaled[0.5]]

Compare with the default map:
GeoGraphics[{EdgeForm[{DotDashed, Blue, Thick}], FaceForm[],
Polygon[("polygon" /. data)[[;; , ;; , 2]]]}, GeoRangePadding -> Scaled[0.5]]

Obtain satellite image of the Eiffel Tower from Google and highlight its basement (in this template the version number of Google maps v=196
may require updating to the current version number):
GeoGraphics[{EdgeForm[{Red, Thickness[.005]}], FaceForm[],
Polygon[("polygon" /. data)[[;; , ;; , 2]]]}, ImageSize -> 1000,
GeoServer -> {StringTemplate[
"http://khms<*Mod[#2+#3,4]*>.googleapis.com/kh?x=`2`&y=`3`&z=`1`&v=196"],
"ZoomRange" -> {1, 21}}]

Obtain satellite image from Google and overlay it with the semi-transparent map from Wikimapia:
satellite =
GeoGraphics[GeoCenter -> GeoPosition[{48.8635510693252, 2.24669540329895}],
GeoServer -> {StringTemplate[
"http://khms<*Mod[#2+#3,4]*>.googleapis.com/kh?x=`2`&y=`3`&z=`1`&v=196"],
"ZoomRange" -> {1, 21}},
GeoRange -> Quantity[2, "Kilometers"], AspectRatio -> 1];
overlay = GeoGraphics[GeoCenter -> GeoPosition[{48.8635510693252, 2.24669540329895}],
GeoServer -> {StringTemplate[
"http://i<*Mod[#2,4]+Mod[#3,4]*4*>.wikimapia.org/?x=`2`&y=`3`&zoom=`1`&lng=0&type=hybrid"],
"ZoomRange" -> {1, 22}}, GeoRange -> Quantity[2, "Kilometers"]];
GraphicsRow[{satellite, overlay}, ImageSize -> 600]

ImageCompose @@ Cases[{satellite, overlay}, _Image, -1]
