Formulas do not allow commas in numbers, probably because commas have a special purpose in formulas: to separate the parameters in functions. While the +-*/ operators could conceivably figure out commas in numbers, it would be inconsistent use of commas. You could end up with something like this mess of a syntax error
=IF(A2=1,234.56+5,432.10, TRUE, FALSE)
Use Paste and Match Style for pasting in a single number without affecting the cell formatting. I redefined the keyboard shortcut on mine to Command Shift V. I don't recall the original shortcut but I believe it involved an extra finger or two and maybe a toe. It is common to a lot of Apple apps so redefine it for "all applications" if you are going to redefine. System Preferences->Keyboard->Keyboard Shortcuts->App Shortcuts or wherever it is in more recent MacOS's than I have. The menu item needs to be typed in EXACTLY as it appears in the menu.
If you need to do what you showed in your last screenshot where you are pasting several numbers into the same cell, a "Service" (Quick Action) can be created in the Automator app that will remove commas from whatever has been copied onto the clipboard. You can then assign a keyboard shortcut to it for ease of use. I suppose this can all be done in the new "Shortcuts" app, too, but I have been unsuccessful assigning a keyboard shortcut to one.

set TempTID to AppleScript's text item delimiters
set AppleScript's text item delimiters to ","
set theText to the clipboard as string
set theTextItems to every text item of theText
set AppleScript's text item delimiters to ""
set theText to theTextItems as string
set AppleScript's text item delimiters to TempTID
set the clipboard to theText
Name it something like 'Remove Commas From Clipboard Text" or something else that makes sense to you. It will show up under the Numbers menu as a Service. If you used "in any application" (like I did) it will show up in other apps, too, like Safari, Pages, etc.. Use "in Numbers" if you want it to be only in Numbers.
Assign a keyboard shortcut to it just like you can do with any other menu item.
Copy your number (Command C), use your keyboard shortcut to remove the comma(s), Paste (Command V)
Note: If you copy a column of cells in Numbers, this script will add an extra cell above and below when pasted back into Numbers. I have no idea how to make it stop that other than rewriting it specifically for Numbers to delete those artifacts. But you're not using it for that purpose anyway.