How to perform time series analysis that contains multiple groups in Python using fbProphet or other models?

fbprophet requires two columns ds and y, so you need to first rename the two columns

df = df.rename(columns={'Date': 'ds', 'Amount':'y'})

Assuming that your groups are independent from each other and you want to get one prediction for each group, you can group the dataframe by "Group" column and run forecast for each group

from fbprophet import Prophet
grouped = df.groupby('Group')
for g in grouped.groups:
    group = grouped.get_group(g)
    m = Prophet()
    m.fit(group)
    future = m.make_future_dataframe(periods=365)
    forecast = m.predict(future)
    print(forecast.tail())

Take note that the input dataframe that you supply in the question is not sufficient for the model because group D only has a single data point. fbprophet's forecast needs at least 2 non-Nan rows.

EDIT: if you want to merge all predictions into one dataframe, the idea is to name the yhat for each observations differently, do pd.merge() in the loop, and then cherry-pick the columns that you need at the end:

final = pd.DataFrame()
for g in grouped.groups:
    group = grouped.get_group(g)
    m = Prophet()
    m.fit(group)
    future = m.make_future_dataframe(periods=365)
    forecast = m.predict(future)    
    forecast = forecast.rename(columns={'yhat': 'yhat_'+g})
    final = pd.merge(final, forecast.set_index('ds'), how='outer', left_index=True, right_index=True)

final = final[['yhat_' + g for g in grouped.groups.keys()]]

import pandas as pd
import numpy as np
from statsmodels.tsa.statespace.sarimax import SARIMAX
from statsmodels.tsa.arima_model import ARIMA
from statsmodels.tsa.stattools import adfuller
from matplotlib import pyplot as plt
from sklearn.metrics import mean_squared_error
from sklearn.metrics import mean_squared_log_error  



# Before doing any modeling using ARIMA or SARIMAS etc Confirm that
# your time-series is stationary by using Augmented Dick Fuller test
# or other tests.

# Create a list of all groups or get from Data using np.unique or other methods
groups_iter = ['A', 'B', 'C', 'D']

dict_org = {}
dict_pred = {}
group_accuracy = {}

# Iterate over all groups and get data 
# from Dataframe by filtering for specific group
for i in range(len(groups_iter)):
    X = data[data['Group'] == groups_iter[i]]['Amount'].values
    size = int(len(X) * 0.70)
    train, test = X[0:size], X[size:len(X)]
    history = [x for in train]

    # Using ARIMA model here you can also do grid search for best parameters
    for t in range(len(test)):
        model = ARIMA(history, order = (5, 1, 0))
        model_fit = model.fit(disp = 0)
        output = model_fit.forecast()
        yhat = output[0]
        predictions.append(yhat)
        obs = test[t]
        history.append(obs)
        print("Predicted:%f, expected:%f" %(yhat, obs))
    error = mean_squared_log_error(test, predictions)
    dict_org.update({groups_iter[i]: test})
    dict_pred.update({group_iter[i]: test})

    print("Group: ", group_iter[i], "Test MSE:%f"% error)
    group_accuracy.update({group_iter[i]: error})
    plt.plot(test)
    plt.plot(predictions, color = 'red')
    plt.show()