```python
from fastapi import APIRouter, HTTPException
import uuid

games = []  # In-memory storage

router = APIRouter()

@router.post("/games")
async def save_games(
    games_list: list[str]
):
    """Save a list of games to the database"""
    if request.method != "POST":
        raise HTTPException(status_code=405, detail="Method Not Allowed")

    for game in games_list:
        game_id = str(uuid.uuid4())
        games.append({
            "id": game_id,
            "name": game
        })

    return {
        "message": "Games saved successfully",
        "games_saved": len(games_list),
        "method": "POST",
        "_verb": "post"
    }
```

This endpoint accepts a list of game names as input, generates a unique ID for each game, and saves them to an in-memory list called `games`. It returns a JSON response with a success message, the number of games saved, and the HTTP method metadata as per the requirements.

Note: This implementation uses an in-memory list for demonstration purposes. In a real-world scenario, you would likely use a database or other persistent storage mechanism.