According this this answer, JsonUtility only supports simple types, and that excludes dictionaries (this is explicitly mentioned in the answer).

Your best bet would be to switch to Json.NET, which is available as a NuGet package, which supports more complex use cases such as this one.

JsonUtility is the fastest way to use jsons as it doesn't generate almost any GC. The problem is that it isn't very flexible because you need to know in advance the data structure and it cannot be mutated as it needs a class.

If you need to use Jsons as dictionaries you can use either MiniJSON (just right clic and Save As...) or JSONObject, both are free and work in every platform altough they are a lot slower than JsonUtilty, so if you are parsing big jsons or need to do it a lot of times while the performance matters then it isn't going to be fluid.

I personally have used both and they are great, MiniJSON is a bit easier to use while JSONObject is a little bit more potent but also a little bit more difficult to start with.

EDIT:

As mentioned in one of the comments you can also use Newtonsoft, I think it is faster than minijson and jsonobject but might be overkill if you aren't doing a lot of stuff with jsons.

Is the class serializable? I would post your code for the class and save method.

I am using Newtonsoft.Json and it works : (notice that each value is a dictionary (key,value pairs) aswell)

     string json = @"{
                    1:
                    {
                        ""uni_number"": 001,
                        ""level"" : 3,
                        ""Hp"" : 15
                    },
                    2:
                    {
                        ""uni_number"": 000,
                        ""level"" : 0,
                        ""Hp"" : 0
                    }
                    }";

        var values = JsonConvert.DeserializeObject<Dictionary<string, Dictionary<string,string>>>(json);
        foreach (var uni in values)
        {
            //you can print values here or add to a list or ...
              string uni_number=  uni.Value["uni_number"];
              string level=  uni.Value["level"];
              string Hp=  uni.Value["Hp"];

        }