More an observation than an answer (I post it here because I cannot put figures in comments).
The behaviour seems to be caused by the `Dashed` attribute, and it is the line, not the point, that is not drawn correctly.

Op's code for me (v11.0.1 x64 on Windows 10) produces

 point1 = {57.7538, 44.0056};
 point2 = {57.7607, 43.9983};
 line1 = Line[{point1, point2}];
 line2 = Line[{{54.3101, 40.715}, {58.4058, 44.624}}];
 intersectionPoint =
 First[{x, y} /.
 FindInstance[{x, y} \[Element] line1 && {x, y} \[Element]
 line2, {x, y}]];
 Graphics[{
 {Gray, line1},
 {PointSize[Large], Point[{point1, point2}]},
 {PointSize[Large], Red, Point[intersectionPoint]}
 },
 Prolog -> {Dashed, line2},
 Frame -> True, ImageSize -> Large
 ]

<img src="https://i.sstatic.net/tTncR.png" width="400">

Removing the `Dashed` attribute in the `Prolog` option, I get:

 point1 = {57.7538, 44.0056};
 point2 = {57.7607, 43.9983};
 line1 = Line[{point1, point2}];
 line2 = Line[{{54.3101, 40.715}, {58.4058, 44.624}}];
 intersectionPoint =
 First[{x, y} /.
 FindInstance[{x, y} \[Element] line1 && {x, y} \[Element]
 line2, {x, y}]];
 Graphics[{
 {Gray, line1},
 {PointSize[Large], Point[{point1, point2}]},
 {PointSize[Large], Red, Point[intersectionPoint]}
 },
 Prolog -> {line2},
 Frame -> True, ImageSize -> Large
 ]

<img src="https://i.sstatic.net/w1Eam.png" width="400">

Drawing `line2` inside the same `Graphics` as the rest of the objects doesn't change this, so it doesn't look like a problem in `Prolog`.

Even better, the lower the `Dashing`, the further away the line is drawn from the intersection point:

 point1 = {57.7538, 44.0056};
 point2 = {57.7607, 43.9983};
 line1 = Line[{point1, point2}];
 line2 = Line[{{54.3101, 40.715}, {58.4058, 44.624}}];
 intersectionPoint =
 First[{x, y} /.
 FindInstance[{x, y} \[Element] line1 && {x, y} \[Element]
 line2, {x, y}]];
 Graphics[{
 {Gray, line1},
 {PointSize[Large], Point[{point1, point2}]},
 {PointSize[Large], Red, Point[intersectionPoint]}
 },
 Prolog -> Map[{Hue[10 #], Dashing@#, line2} &]@Range[0, .5, 0.01],
 Frame -> True, ImageSize -> Large
 ]

<img src="https://i.sstatic.net/erozt.png" width="500">

I have no idea why this happens.