[Fizzlemizz]

I mean't that in your original code wherever you had

TestMMBicon:Hide("TestMiniMapButton")

you needed to add

self.db.show = false

and wherever you had

TestMMBicon:Show("TestMiniMapButton")

you needed to add

self.db.show = true

Probably a bit more like:

local TestMMBicon = LibStub("LibDBIcon-1.0", true)
local TestMiniButton = LibStub("AceAddon-3.0"):NewAddon("TestMiniMapButton", "AceConsole-3.0")
 
SLASH_MMBSHOW1, SLASH_MMBSHOW2 = "/mnb", "/MNB";
function SlashCmdList.MMBSHOW(msg, editbox)
  TestMMBicon:Show("TestMiniMapButton")
  TestMiniButton.db.show = true -- Added
end
 
 
SLASH_MMBHIDE1, SLASH_MMBHIDE2 = "/mnbh", "/MNBH";
function SlashCmdList.MMBHIDE(msg, editbox)
  TestMMBicon:Hide("TestMiniMapButton")
  TestMiniButton.db.show = false -- Added
end
 
 
local miniButton = {
  text = "TestMiniMapButton",
  type = "data source",
  icon = "Interface/Minimap/Vehicle-AllianceWarlockPortal",
  OnTooltipShow = function(tooltip)
  if not tooltip or not tooltip.AddLine then return end
  tooltip:AddLine("|cffff0000Map|r|cff00ccffNotes|r")
  end,
  OnClick = function(self, button)
    if button == "RightButton" then 
      LibStub("AceConfigDialog-3.0"):Close("TestMiniMapButton")
    end
    if button == "LeftButton" then
      LibStub("AceConfigDialog-3.0"):Open("TestMiniMapButton")
    end
    if IsShiftKeyDown() and button == "RightButton" then
      TestMMBicon:Hide("TestMiniMapButton")
    end
end}
 
 
function TestMiniButton:OnInitialize()
  self.db = LibStub("AceDB-3.0"):New("TestMiniMapButtonDB", { profile = { minimap = { hide = false, }, }, })
  TestMMBicon:Register("TestMiniMapButton", miniButton, self.db.profile.minimap)
end
 
 
function Addon:PLAYER_LOGIN()
local options = {
  type = "group",
  name = "Test",
  childGroups = "tab",
  desc = "test",
  get = function(info) return db[info[#info]] end,
  set = function(info, v) db[info[#info]] = v HandyNotes:SendMessage("HandyNotes_NotifyUpdate", "TestAddon") end,
  args = {  
    GeneralTab = {
      type = "group",
      name = "General",
      desc = "General settings that apply to Azeroth / Continent / Dungeon map at the same time",
      order = 0,
      args = {
        hideMapNotesMMB = {
          type = "header",
          name = "-> MiniMapButton <-",
          order = 1,
          },
        showMMB = {
          type = "execute",
          name = L["show"],
          desc = L["Show the minimap button on the minimap"],
          order = 1.1,
          width = 1.89,
          get = function() return db.show["ShowMMB"] end,
          func = function() 
            TestMMBicon:Show("TestMiniMapButton") 
            TestMiniButton.db.show = true -- Added
          end
          },  
        hideMMB = {
          type = "execute",
          name = L["hide"],
          desc = L["Hide the minimap button on the minimap"],
          order = 1.2,
          width = 1.89,
          get = function() return db.show["HideMMB"] end,
          func = function() 
            TestMMBicon:Hide("TestMiniMapButton") 
            TestMiniButton.db.show = false -- Added
          end
          },
      }
    }
  }
}
 
  HandyNotes:RegisterPluginDB("TestAddon", pluginHandler, options)
  self.db = LibStub("AceDB-3.0"):New("HandyNotes_TestAddonDB", defaults, true)
  db = self.db.profile
  LibStub("AceConfigRegistry-3.0"):RegisterOptionsTable("TestMiniMapButton", options)
  Addon:RegisterEvent("PLAYER_ENTERING_WORLD")
end

EDIT: Because Ace doesn't have a SetShown method.
At PLAYER_LOGIN you would add:

if not db.show then
      TestMMBicon:Hide("TestMiniMapButton")
end

to set the initial state of the button when the user logs in to what was last saved in the db.