Without knowing much about the libraries you are using I can see you are creating these annotation objects and then assigning them to a global variable that is re-assigned later and thus you lose the right object to make it visible.

Instead you could keep the annotation objects into a dictionary and try to retrieve them later when you need them based on an object.

I used a list to show you the idea, but you need a dictionary I guess to identify the right objects.

I modified your code a bit and it shows the desired behaviour if you resize the window...I guess you have to find a way to refresh the plot also:

import pandas as pd
import numpy as np
import matplotlib as mtpl
from matplotlib import pyplot as plt
import matplotlib.ticker as ticker

annotations = []
data = {
    # 'Name': ['Status', 'Status', 'HMI', 'Allst', 'Drvr', 'CurrTUBand', 'RUSource', 'RUReqstrPriority', 'RUReqstrSystem', 'RUResReqstStat', 'CurrTUBand', 'DSP', 'SetDSP', 'SetDSP', 'DSP', 'RUSource', 'RUReqstrPriority', 'RUReqstrSystem', 'RUResReqstStat', 'Status', 'Delay', 'Status', 'Delay', 'HMI', 'Status', 'Status', 'HMI', 'DSP'],
    # 'Value': [4, 4, 2, 1, 1, 1, 0, 7, 0, 4, 1, 1, 3, 0, 3, 0, 7, 0, 4, 1, 0, 1, 0, 1, 4, 4, 2, 3],
    # 'Gage': ['H1', 'H3', 'H3', 'H3', 'H3', 'H3', 'H3', 'H3', 'H3', 'H3', 'H3', 'H3', 'H3', 'H3', 'H3', 'H3', 'H3', 'H3', 'H3', 'H1', 'H1', 'H3', 'H3', 'H3', 'H1', 'H3', 'H3', 'H3'],
    # 'Id_Par': [0, 0, 0, 0, 0, 0, 10, 10, 10, 10, 10, 0, 0, 22, 22, 28, 28, 28, 28, 0, 0, 38, 38, 0, 0, 0, 0, 0]
    'Name': ['Lamp_D_Rq', 'Status', 'Status', 'HMI', 'Lck_D_RqDrv3', 'Lck_D_RqDrv3', 'Lck_D_RqDrv3', 'Lck_D_RqDrv3', 'Lamp_D_Rq', 'Lamp_D_Rq', 'Lamp_D_Rq', 'Lamp_D_Rq'],
    'Value': [0, 4, 4, 2, 1, 1, 2, 2, 1, 1, 3, 3],
    'Gage': ['F1', 'H1', 'H3', 'H3', 'H3', 'F1', 'H3', 'F1', 'F1', 'H3', 'F1', 'H3'],
    'Id_Par': [0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0]
    }

signals_df = pd.DataFrame(data)


def plot_signals(signals_df):
    print(signals_df)
    # Count signals by parallel
    signals_df['Count'] = signals_df.groupby('Id_Par').cumcount().add(1).mask(signals_df['Id_Par'].eq(0), 0)
    # Subtract Parallel values from the index column
    signals_df['Sub'] = signals_df.index - signals_df['Count']
    id_par_prev = signals_df['Id_Par'].unique()
    id_par = np.delete(id_par_prev, 0)
    signals_df['Prev'] = [1 if x in id_par else 0 for x in signals_df['Id_Par']]
    signals_df['Final'] = signals_df['Prev'] + signals_df['Sub']
    # Convert and set Subtract to index
    signals_df.set_index('Final', inplace=True)

    # Get individual names and variables for the chart
    names_list = [name for name in signals_df['Name'].unique()]
    num_names_list = len(names_list)
    num_axisx = len(signals_df["Name"])

    # Matplotlib's categorical feature to convert x-axis values to string
    x_values = [-1, ]
    x_values += (list(set(signals_df.index)))
    x_values = [str(i) for i in sorted(x_values)]

    # Creation Graphics
    fig, ax = plt.subplots(nrows=num_names_list, figsize=(10, 10), sharex=True)
    plt.xticks(np.arange(0, num_axisx), color='SteelBlue', fontweight='bold')

    # Loop to build the different graphs
    for pos, name in enumerate(names_list):
        print("name: %s" % name)
        print("pos: %s" % pos)
        # Creating a dummy plot and then remove it
        dummy, = ax[pos].plot(x_values, np.zeros_like(x_values))
        dummy.remove()

        # Get names by values and gage data
        data = signals_df[signals_df["Name"] == name]["Value"]
        data_gage = signals_df[signals_df["Name"] == name]["Gage"]

        # Get values axis-x and axis-y
        x_ = np.hstack([-1, data.index.values, len(signals_df) - 1])
        y_ = np.hstack([0, data.values, data.iloc[-1]])
        y_gage = np.hstack(["", "-", data_gage.values])
        # print(y_gage)

        # Plotting the data by position
        steps = ax[pos].plot(x_.astype('str'), y_, drawstyle='steps-post', marker='*', markersize=8, color='k', linewidth=2)
        ax[pos].set_ylabel(name, fontsize=8, fontweight='bold', color='SteelBlue', rotation=30, labelpad=35)
        ax[pos].yaxis.set_major_formatter(ticker.FormatStrFormatter('%0.1f'))
        ax[pos].yaxis.set_tick_params(labelsize=6)
        ax[pos].grid(alpha=0.4, color='SteelBlue')
        # Labeling the markers with Values and Gage
        xy_temp = []
        for i in range(len(y_)):
            if i == 0:
                xy = [x_[0].astype('str'), y_[0]]
                xy_temp.append(xy)
            else:
                xy = [x_[i - 1].astype('str'), y_[i - 1]]
                xy_temp.append(xy)

            # Creating values in text inside the plot
            ax[pos].text(x=xy[0], y=xy[1], s=str(xy[1]), color='k', fontweight='bold', fontsize=12)

            for val_gage, xy in zip(y_gage, xy_temp):
                print("val_gage: %s" % val_gage)
                annot = ax[pos].annotate(val_gage, xy=xy, xytext=(-20, 20), textcoords="offset points",
                                         bbox=dict(boxstyle="round", fc="w"),
                                         arrowprops=dict(arrowstyle="->"))

                annot.set_visible(False)
                annotations.append(annot)

    # Function for storing and showing the clicked values
    def update_annot(ind):
        print("Enter update_annot")
        coord = steps[0].get_offsets()[ind["ind"][0]]
        annot.xy = coord
        text = "{}, {}".format(" ".join(list(map(str, ind["ind"]))),
                                " ".join([y_gage[n] for n in ind["ind"]]))
        annot.set_text(text)
        annot.get_bbox_patch().set_alpha(0.4)

    def on_click(event):
        print("Enter on_click")
        vis = annot.get_visible()
        # make the first three annotations visible
        for i in range(0, 3):
            print('elem visible')
            annotations[i].set_visible(True)
        print(event.inaxes)
        print(ax[pos])
        print(event.inaxes == ax[pos])
        if event.inaxes == ax[pos]:
            cont, ind = steps[0].contains(event)
            print (ind)
            if cont:
                update_annot(ind)
                annot.set_visible(True)
                fig.canvas.draw_idle()
            else:
                if vis:
                    annot.set_visible(False)
                    fig.canvas.draw_idle()

    fig.canvas.mpl_connect("button_press_event",on_click)

    plt.show()

plot_signals(signals_df)

I hope this helps and it will solve your problem. It looks more like a python/programming problem and not so much related to the libraries you are using if I get this right :)

Using Plotly for data annotation labels animation upon mouse hovering over graph data points

Not to mention a huge slew of other awesome, easy-to-use, widely-compatible JS-interactive graphing capabilities, all free, all in Python. Just install with conda (or pip) no online account required and the plots default to "offline mode" in latest version(s).

So with plotly, specifically plotly express, it's very simple!

I am not 100% what you want as far as specifics for your axes/data, but I think below demonstrates the tremendous ease with which Plotly can be used to create interactive graphs, & the very powerful customization available.

You will easily be able to adapt these interactive graphs to your desired purposes via cursory perusal through the plotly docs.

And through plotly.express you still have access to the built-in Fig features relevant to all the other submodules, too. So don't overlook those [e.g., the docs link above shows sections specific for subplotting, custom annnotations/hover annotations, custom style formatting, etc., all which still apply to objects within plotly.express!]).

I - Data Structures Setup

Identical to yours... Plotly is designed to work with pandas.DataFrames, specifically*.

E.g.,

import plotly.express as px
import plotly.graph_objs as go

import pandas as pd
import numpy as np

data = {
    "Name": [
        "Lamp_D_Rq", "Status", "Status", "HMI",
        "Lck_D_RqDrv3", "Lck_D_RqDrv3", "Lck_D_RqDrv3",
        "Lck_D_RqDrv3", "Lamp_D_Rq", "Lamp_D_Rq",
        "Lamp_D_Rq", "Lamp_D_Rq",
    ],
    "Value": [0, 4, 4, 2, 1, 1, 2, 2, 1, 1, 3, 3],
    "Gage": [
        "F1", "H1", "H3", "H3", "H3",
        "F1", "H3", "F1", "F1", "H3",
        "F1", "H3",
    ],
    "Id_Par": [0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0],
}

signals_df = pd.DataFrame(data)

NOTE: I then ran signals_df through your plotting function, and added return signals_df to get the updated df, which was:

II - Plotting custom hover annotations with plotly.express (px)

Here's one relatively (i.e., to mpl) quite simple, possible multi-featured, modern interactive display of your data using Plotly (via px):

fig = px.line(
    signals_df,
    y="Value",
    x="Sub",
    color="Name",
    hover_data=["Gage"],
    custom_data=["Gage"],
    markers=True,
    height=500,
    render_mode="svg")

fig.update_traces(line={"shape": 'hv'})
fig.update_traces(
    hovertemplate="<br>".join([
        "Gage: %{customdata[0]}",
    ])
)
fig.show(config={'displaylogo': False})