Implicating tamamo_no_mae_(fate) → fate_(series) fate/extra fate/grand_order
Link to implication

Reason:

The character is from these games.

No. Should just imply Fate/Extra, her game of origin, which should imply the series tags.

And we use Caster_(Fate/Extra), anyways. Right now, at least.

I would suggest an alias of

tamamo_Cat -> tamamo_no_mae_(fate)
caster_(fate/extra) -> tamamo_no_mae_(fate)

Caster_(fate/extra) seems like a name that originated during the time when the characters actual name wasn't known, considering how every other fate character has a name beyond their class.

Well, within the story itself Servants are often called just their class, even when people know their names. (Only Gilgamesh is regularly called anything but his class in Fate/stay night).

Tamamo Cat shouldn't imply Tamamo. They're separate beings, just beings that splintered off from the same original.