Observation:

Installation instructions are at the bottom of this page.

Snapping settings for 1 selected object

Move Tool

When moving the 3D cursor, the Gizmo can be moved. So, just choose where want to move the object; click on the cursor/gizmo and move the object with snap to the point where the cursor/gizmo was before starting the movement. These movements can be done with just the left mouse button, without any extra hotkeys.

Rotate & Scale Tools (Closest)

For Rotate and Scale choose the rotation pivot with snap target "Closest". I believe that in most cases, the “Closest” snap target is sufficient here.

Rotate & Scale Tools (Target)

If need of the snap in a more complex point, move the cursor/gizmo to that point and send the origin to there, using “Set Target”; which is actually “Origin to 3D Cursor”. I renamed this function to match the purpose of this add-on.

Snapping settings for 2 selected objects

Move Tool

In addition to the two other previous options; When 2 objects are selected, two new snapping options appear, and they are: 

Pivot Point “Median” + Snap with “Closest”; and 

Pivot Point “Median” + Snap with “Median”

For this context, no other options are necessary.

Rotate & Scale Tools

For Rotate, the Rotate + “Target” pivot point is the same as Rotate + “Median” (using snap with “Median” was necessary, as snap with “Active” does not allow you to move the Gizmo precisely). In addition to these previous options, two new options are available:

Rotate:

Pivot Point “Median” + Snap with “Active”; and

Pivot Point “Individual origins” + Snap with “Active”

Scale:

Pivot Point “Median” + Snap with “Closest”; It is

Pivot Point “Individual origins” + Snap with “Active”.

Snapping settings for 3 selected objects (or more)

A new snap option for this context is: Pivot Point “Bounding Box Center” + Snap with “Center”. This pivot point only works from 3 selections or more. For Rotate and Scale the target is "Active".

The criteria followed for the Move tool is: “grab the object by the pivot and snap it to the same pivot”; with the exception of the snap with: "Closest".

The pivot changes when the bounding box changes position with each rotation. For Scale the scaling pivot is not change.

The pivot changes when the bounding box changes position with each rotation. For Scale the scaling pivot is not change.

Other functions

Cursor orientation mode

The “Cursor” orientation mode is very useful for aligning and mirroring objects (that are outside the X, Y and Z global and local axes) to any direction, based on the Cursor orientation. By default, orienting the Cursor to the object's geometry is not very practical; as the accessibility of the tools necessary for this makes this task very laborious. In this case here, position the Cursor/Gizmo to the geometry of any object using “Align Gizmo”, which is the “Align Rotation to Target” function. This allows objects to be aligned with just a few movements, aligning the Rotate tool Gizmo to the surface of the objects to be aligned.

Mirror to Z axis

Flip the object on the “Z” axis in any direction using the “Move” tool Gizmo with “Align Gizmo”.

Align objects to Cursor orientation

In the “Cursor” orientation mode there are two tools called “Align to”; which aligns the selected object to the Cursor orientation and, “Align/Send to”; which does the same thing; however, it sends the object to the Cursor position. Align the Cursor/Gizmo and give zoom in wherever you want and press the “G” key; then the object will be taken to that specific location. If “Align Gizmo” is active, the object will be aligned according to the Cursor/Gizmo position; so be sure to align it before each movement.

Snap on instances

Move the instances without needing to move the origin. In the case of Rotate and Scale, use “Rotate+Closest”. Using “Set Target” will move the origin and throw off the object’s coordinates.

Assign shortcuts individually to each tool

If want, use custom shortcuts for each tool. I used the default shortcuts for the Move, Rotate and Scale tools as an example (make sure to disable the shortcut already used).

In this case, I used the main Move tool (to move the Cursor/Gizmo) and the Rotate and Scale + “Closest” tools.

This is a list of all addresses of the Move, Rotate and Scale tools, considering all contexts and selections.

Snapping settings for 1, 2 and 3 selected elements (or more)

Works as in object mode; bearing in mind that the active element is equivalent to the origin of the active object, and the number of selected elements is equivalent to the number of selected objects; so that the right snapping combinations match the selection context.

Move, Rotate & Scale Tools

Moving with Gizmo alignment

In “Cursor” orientation mode, Use “Align Gizmo” to move the vertex with the Cursor/Gizmo following the geometry orientation.

Rotate

In the case of Rotate, the available option is useful for transform just one vertex. The same for the Scale tool, however, it didn't seem to make much sense in this context, but it is an available option.

Supported modes

Some modes have tools in common, so the add-on operates in the following modes:
Object Mode; Edit Mode; Edit Curve; Surface Mode; Metaball Mode; Armature Mode; Pose Mode, and Lattice Mode.
However, the focus of this add-on is Object Mode and Edit Mode. The other modes are optional.

Inactivity alert

If there is no object or element selected, a warning will appear in place of the tools that require a selection. The other tools (whose activation does not depend on a selection) will continue to be visible.

Icon Theme

Change the icon theme to “dark” or “light” to suit your preferred theme.

Installation

Install the “native_snapping.zip” file like any other add-on.

Assign a shortcut

Easily choose a shortcut for the add-on:
In Keymap, access:
3D View > Object Mode >
                  Object Mode (Global)
                  Mesh > Mesh (Global)
Click add new.
Then, in 'none' look for the identifier:
wm.call_menu_pie (with the characters: _pie at the end)
Then, in 'Name', also look for the Class:
PIE_MT_NativeSnapping
Finally, add a shortcut of your choice.

For the other modes, the procedure for assign a shortcut is the same.