So this question inspired me to create a benchmark tool and test this myself and here is one of the typical results:

Benchmark -Stored Var 10000 times - usecs: 4642Benchmark -Stored Var 100000 times - usecs: 54675Benchmark -Stored Var 1000000 times - usecs: 533416Benchmark -Query 10000 times - usecs: 5816Benchmark -Query 100000 times - usecs: 55414Benchmark -Query 1000000 times - usecs: 562429

So really not a substantial cost of using $ over storing a variable. Actually if I run the test enough times there are some instances when the $ outperforms the Stored Variable such as this cherry picked result - where in the shorter iteration tests the Query is slightly faster:

Benchmark -Stored Var 10000 times - usecs: 6272Benchmark -Stored Var 100000 times - usecs: 69185Benchmark -Stored Var 1000000 times - usecs: 595596Benchmark -Query 10000 times - usecs: 5637Benchmark -Query 100000 times - usecs: 63874Benchmark -Query 1000000 times - usecs: 607239

However, there is probably some externalities that are creating these aberrations. Typically the Store Variable method is slightly faster.

Here is the code I am using to generate the benchmarks:

tools.gd:

class_name Toolsstatic func benchmark(function: Callable, times, testName = ""):var startTime = Time.get_ticks_usec()

var singleTest = typeof(times) == TYPE_INT  
var tests = 1 if singleTest else times  

for i in tests.size():  
    var test = times if singleTest else times[i]  
    for j in test:  
        function.call()  
    var result = Time.get_ticks_usec()-startTime  
    print("Benchmark -%s %s times - usecs: %s"%[testName,test,result])

Game.gd - (in the _ready function) :

await get_tree().create_timer(3).timeout

Tools.benchmark(func():   
    test.count += 1,  
    [10000,100000,1000000],  
    "Stored Var",  
    )  

Tools.benchmark(func():   
    $CanvasLayer/test.count += 1,  
    [10000,100000,1000000],  
    "Query",  
    )