Is it possible to expose the same Swagger JSON in both Swagger 2.0 and Open API 3 formats with Swashbuckle in ASP .NET Core?

An alternative approach is to split the request pipeline:

// serve v3 from /swagger/v1/swagger.json
app.UseSwagger(o => o.RouteTemplate = "swagger/{documentName}/swagger.json");

// serve v2 from /swagger2/v1/swagger.json
app.Map("/swagger2", swaggerApp => 
    swaggerApp.UseSwagger(options => {
        // note the dropped prefix "swagger/"
        options.RouteTemplate = "{documentName}/swagger.json";
        options.SerializeAsV2 = true;
    })
);

You can serialize the OpenAPI document as V2 and serve it yourself. Taking SwaggerMiddleware as reference:

First register SwaggerGenerator:

services.AddTransient<SwaggerGenerator>();

Then inject SwaggerGenerator and build the document. Serve it from an endpoint or a controller. You can take the version as a path parameter to decide which version to serve.

app.UseEndpoints(e =>
{
    // ...
    e.MapGet("/swagger/{documentName}/swagger.{version}.json", context =>
    {
        var documentName = context.Request.RouteValues.GetValueOrDefault("documentName", null) as string;
        var version = context.Request.RouteValues.GetValueOrDefault("version", null) as string;
        if (documentName is null || version is null)
        {
            context.Response.StatusCode = StatusCodes.Status400BadRequest;
            return Task.CompletedTask;
        }

        // inject SwaggerGenerator
        var swaggerGenerator = context.RequestServices.GetRequiredService<SwaggerGenerator>();
        var doc = swaggerGenerator.GetSwagger(documentName);

        // serialize the document as json
        using var writer = new StringWriter(CultureInfo.InvariantCulture);
        var serializer = new OpenApiJsonWriter(writer);
        if (version == "v2")
        {
            doc.SerializeAsV2(serializer);
        }
        else
        {
            doc.SerializeAsV3(serializer);
        }
        var json = writer.ToString();

        // serve it as json
        context.Response.ContentType = MediaTypeNames.Application.Json;
        return context.Response.WriteAsync(json, new UTF8Encoding(false));
    });
});

When you visit /swagger/v1/openapi.v2.json, you'll get the OpenAPI doc serialized as v2.

{
  "swagger": "2.0",
  "info": {
    "title": "ApiPlayground",
    "version": "1.0"
  },
  "paths": { ... }
}

Whereas /swagger/v1/openapi.v3.json will give you the doc serialized as v3:

{
  "openapi": "3.0.1",
  "info": {
    "title": "ApiPlayground",
    "version": "1.0"
  },
  "paths": { ... },
  "components": { ... }
}