Google Photos tries to recognize faces in photos, and makes groups of similar faces. That process is automatic, but all users can remove photos from a group, and in some regions there is also the option to correct or add tags manually.
The following instructions are primarily written for the Google Photos web app, but similar options are available for mobile devices.
Web app: when you hover with the mouse over the name of a group, a pencil does appear. Clicking on the pencil allows to change the name. Confirm by clicking on "Done".
On mobile: Tap on the Name to get in Edit mode.
You can label your face group as "me". See https://support.google.com/photos/answer/7378566
You can remove photos from a group for several reasons:
not a person
wrong person (users having the option to edit face tags can add the photo to the correct group instead)
not the main subject
blurry or poor quality
offensive
To remove photos from a group:
Select the photo(s) in the group (the option is not available when the photo is opened)
Go to 3-dots > Remove results
The photo will be removed from this group, and hopefully added to another group.
Manual tagging of faces is not available for all accounts. For example accounts for which face grouping is activated after it became available in Europe seem to miss this option.
You can see whether this option is available when there is an "Edit" icon (pencil) to the right of "People" in the info of a photo.
To add a photo to a group:
Open the photo and the info panel. You should now see the faces that are added to a group, and possibly also the text "n faces available to add"
Click on the pencil. You should now see all the recognized faces. Hovering over the faces in the photo will highlight the face in the info, and hovering over a face in the info will highlight it in the photo. You can now:
Click on a face with a minus in red if you want to remove or correct > select the correct group or click on + in the right upper corner of the screen to create a new group. Select the correct group. You can use the search function to help you. Recently used groups will appear on top, which makes it easier to tag more photos of the same person.
Click on a face with a + in blue if you want to add it to an existing or new group (as described above)
See https://support.google.com/photos/answer/6128838 under "Combine face groups":
If the same person is in more than one group, you can merge them.
Label one of the face groups with a name or nickname.
Label the other face group with the same name or nickname, choosing from the suggestions.
When you confirm the second name or nickname, Google Photos will ask if you want to merge the face groups.
You might get a suggestion to merge 2 face groups. If they’re the same person, click Yes.
It happens that Google does not recognize a person or pet. Unfortunately you cannot draw a box to add it to the group. If you do not want to loose the information, you can add the name as Description, or add the photo to an album you create(d) for that person/pet.
Google photos may not be able to distinguish between similar pets, twins or siblings. In some regions you can "Edit" the face tags, in other (like Europe) you can only remove wrong results. As an alternative you can create a "normal" album for some people/pets.
When you see an "Edit" icon (pencil) to the right of People/pets in the info of a photo, you can add detected faces to a group or change wrongly grouped faces to the correct group. But unfortunately the edit option is not available in all regions, for example in Europe it seems missing from accounts where Face grouping was activated after it was rolled out in Europe.
When "Edit" is available, you can also create "Live albums", where photos for selected people/pets are automatically added. Advantage of creating your own albums is that corrections can be made in batch on selected photos. You can also create a new "Live album" starting from a given date as follows:
Create a new album
Do NOT click on "Select people or pets", but on 3-dots > Options
You can now select people or pets, but existing photos are by default excluded (see option at the upper right side of the screen)
When editing is not available you can select the wrong photos, tap 3-dots > Remove results, hoping that Google will learn from it. But when there are many, or when two people do appear in the same photo, you have a problem. Creating your own albums may be the better solution.
Even when Edit is available, it's laborious to correct wrongly grouped faces/pets, especially when there are many. It may be easier to create a normal album for one or both people/pets, add the photos you want and remove the wrong ones from the wrong group.
When a deceased pet is replaced by a similar one, and Google can't distinguish between pet A and B:
You can change the name in A and B. Till a give date it shown photos of pet A, later of pet B.
If you really want to separate them, you can:
Change the name of the actual group to pet B, because Google keeps adding photos of pet B to this group
Select all photos of pet A (= till a given date). See how to Select multiple pictures
Add these photos to a new album, and give it the title pet A.
Select the photos of pet A again, tap or click 3-dots > Remove results.
That leaves the group with only photos of pet B
Optional: create a "Live album" starting with a given date for pet B (see above).
If you do not want to see photos of a passed-away pet/person in memories, you can "archive" these photos, which means that they are moved from the Photos tab to Archive. Archived photos are not used for memories, but the photos stay in albums.