There are many ways to get the area of a geometry. I leave here a summary.
$area expression returns the ellipsoidal area (if there is a defined ellipsoid) in the units defined in the Project Properties, in the Measurements section of the General tab.
The Measure Area tool, takes the coordinates from the canvas, but if an ellipsoid is selected in the Project Properties, General, Measurement section, it converts the coordinates to that ellipsoid and calculates the ellipsoidal areas. You can change the units for the tool.
area($geometry) expression, returns the planimetric area, calculated through the coordinates of the geometries in their layer CRS, with the units of their layer CRS.
With the Identify tool, you can get both, ellipsoidal (based on project ellipsoid) and planimetric (based on layer CRS) areas for a feature geometry.
You can set None / Planimetric in the Ellipsoid field of the Project Properties, General tab, Measurement section.
In that case, the
$area function and the Measurement Area tool will return the planimetric area calculated from the canvas coordinates. The Identify tool will no return an ellipsoidal area.
Canvas coordinates are based on the Project CRS (Project Properties, CRS tab). If the project don't have a CRS defined, you will see in the canvas rendered the layers in their original coordinates (without on the fly reprojection).
In Settings menu, Options, CRS tab you can define a defaul CRS for new projects, but you can change the project (canvas) CRS when you want trough the Project Properties, CRS tab.
Also, you can define a default CRS to assign to new layers when they have not a valid CRS already defined (I recommend you the Prompt CRS option, to find out when a layer without CRS is loaded).
The coordinates of a feature geometry in a layer, are defined in the layer CRS. Changing the layer CRS don't change the geometries coordinates. Reprojecting (or Exporting to a different CRS of) the layer change them.