Default styles for Exporting to PDF, EPS and SVG

According to the Documentation,

DefaultStyleDefinitions is a global option that specifies the default stylesheet for all new notebooks.

By spelunking Export with GeneralUtilities`PrintDefinitionsLocal

Needs["GeneralUtilities`"]
PrintDefinitionsLocal[Export]

one can find that for general expr Export first converts it into a Notebook using System`Convert`PDFDump`createVectorExportPacketExpr:

System`Convert`PDFDump`createVectorExportPacketExpr[expr]

Notebook[{Cell[BoxData["expr"], "Output", ShowCellBracket -> False, 
   CellMargins -> {{0, 0}, {0, 0}}]}]

As one can see, this Notebook has no StyleDefinitions option what means that it has to inherit it from the global DefaultStyleDefinitions.

Hence default stylesheet used by Export is determined by DefaultStyleDefinitions option of $FrontEndSession.

---

Changing the default stylesheet

At first in the user profile directory I create custom stylesheet "DefaultMy.nb" based on "Default.nb" but with an additional style named "My":

dir = FileNameJoin[{$UserBaseDirectory, "SystemFiles", "FrontEnd", "StyleSheets"}];
If[! DirectoryQ[dir], CreateDirectory[dir]];
Export[FileNameJoin[{dir, "DefaultMy.nb"}], 
 Notebook[{Cell[StyleData[StyleDefinitions -> "Default.nb"]], 
   Cell[StyleData["My"], FontColor -> Red, FontWeight -> Bold, FontSize -> 250, 
    FontVariations -> {"Underline" -> True}, FontFamily -> "Algerian"]}, 
  StyleDefinitions -> "PrivateStylesheetFormatting.nb"]]

Now I set it as the value for DefaultStyleDefinitions:

CurrentValue[$FrontEndSession, DefaultStyleDefinitions] = "DefaultMy.nb"

After evaluating this "DefaultMy.nb" will be immediately opened as a hidden Notebook what can be verified with the code provided by MB1965:

Cases[FrontEndExecute@FrontEnd`ObjectChildren@$FrontEnd, 
 obj_ /; StringMatchQ[WindowTitle /. AbsoluteOptions[obj, WindowTitle], 
    "Default" ~~ __] :> {obj, WindowTitle /. AbsoluteOptions[obj, WindowTitle]}]    

{{NotebookObject[FrontEndObject[LinkObject["ca2gp_shm", 3, 1]], 12], "DefaultMy.nb"}, 
 {NotebookObject[FrontEndObject[LinkObject["ca2gp_shm", 3, 1]], 4], "Default.nb"}}

If during current front end session there weren't invocations of Export to vector formats, this change will have immediate effect on Export. In the opposite case one should force FrontEnd to do something that will cause the style definition cascade to update: for example, an invocation of Export

ExportString["", "MGF"];

or "ToggleShowExpression" front end token (even when no Cell is selected)

FrontEndTokenExecute["ToggleShowExpression"]

does this. Other methods are discussed here.

Now

text = Style["Algerian", 50, "My"];
ImportString[ExportString[text, "PDF"], "PDF"][[1]]

So I think this resolves at the CurrentValue[$FrontEndSession, DefaultStyleDefinitions] level as can be seen by first finding the notebook object that tracks these styles and inserting your definition there:

defaultStylingNotebook =
  First@
   Select[
    FrontEndExecute@FrontEnd`ObjectChildren@$FrontEnd,
    (WindowTitle /. AbsoluteOptions[#, WindowTitle]) == 
      CurrentValue[$FrontEndSession, DefaultStyleDefinitions] &
    ];

NotebookWrite[defaultStylingNotebook,
 Cell[StyleData["My"], FontColor -> Red, FontWeight -> Bold, 
  FontSize -> 20, FontFamily -> "Times"]
 ]

Now running this:

text = Style["Text", "My"];

Export[
  FileNameJoin@{$TemporaryDirectory, "temp.pdf"},
  text
  ] // Import

I get the red import one would expect:

If you want a work around without doing anything so drastic as this, try this:

text=Style["Text",CurrentValue[EvaluationNotebook[],{StyleDefinitions,"My"}]]