How can I make a stylesheet where inline cells are styled like input cells?

It looked like I had misunderstood "syntax highlighting" in this context. What I need turned out to be ShowCodeAssist->True. Combining this and settings on font and others that looked relevant:

Cell[StyleData["InlineCell"],
ShowAutoStyles->True, 
ShowCodeAssist->True,
LanguageCategory->"Mathematica",
FormatType->"InputForm",
FontFamily->"Courier New", 
FontWeight->"Bold"]

I finally achieved what was almost what I was looking for.

You can change the DefaultInlineFormatType for "Text" cells to DefaultInputInlineFormatType (i.e., "StandardForm"), and change the style "InlineCell" to inherit options from "Input":

SetOptions[
    EvaluationNotebook[],
    StyleDefinitions -> Notebook[
        {
        Cell[StyleData[StyleDefinitions -> "Default.nb"]],
        Cell[StyleData["Text"], DefaultInlineFormatType -> System`DefaultInputInlineFormatType],
        Cell[StyleData["InlineCell", StyleDefinitions->StyleData["Input"]]]
        }
    ]
]

Here is screen capture from a notebook, showing a text cell with an inline cell, followed by a normal "Input" cell, so that you can compare them: