This is my answer:

def plot_bargraph_with_groupings(df, groupby, colourby, title, xlabel, ylabel):
    """
    Plots a dataframe showing the frequency of datapoints grouped by one column and coloured by another.
    df : dataframe
    groupby: the column to groupby
    colourby: the column to color by
    title: the graph title
    xlabel: the x label,
    ylabel: the y label
    """

    import matplotlib.patches as mpatches

    # Makes a mapping from the unique colourby column items to a random color.
    ind_col_map = {x:y for x, y in zip(df[colourby].unique(),
                               [plt.cm.Paired(np.arange(len(df[colourby].unique())))][0])}


    # Find when the indicies of the soon to be bar graphs colors.
    unique_comb = df[[groupby, colourby]].drop_duplicates()
    name_ind_map = {x:y for x, y in zip(unique_comb[groupby], unique_comb[colourby])}
    c = df[groupby].value_counts().index.map(lambda x: ind_col_map[name_ind_map[x]])

    # Makes the bargraph.
    ax = df[groupby].value_counts().plot(kind='bar',
                                         figsize=FIG_SIZE,
                                         title=title,
                                         color=[c.values])
    # Makes a legend using the ind_col_map
    legend_list = []
    for key in ind_col_map.keys():
        legend_list.append(mpatches.Patch(color=ind_col_map[key], label=key))

    # display the graph.
    plt.legend(handles=legend_list)
    ax.set_xlabel(xlabel)
    ax.set_ylabel(ylabel)

Use seaborn.countplot

import seaborn as sns
sns.set(style="darkgrid")
titanic = sns.load_dataset("titanic")
ax = sns.countplot(x="class", data=titanic)

Ref the documentation of seaborn https://seaborn.pydata.org/generated/seaborn.countplot.html

It might be a little bit too complicated but this does the work. I first defined the mappings from name to industry and from industry to color (it seems like there are only two industries but you can adjust the dictionary to your case):

ind_col_map = {
    "industry1": "red",
    "industry2": "blue"
}

unique_comb = df[["Name","industry"]].drop_duplicates()
name_ind_map = {x:y for x, y in zip(unique_comb["Name"],unique_comb["industry"])}

Then the color can be generated by using the above two mappings:

c = df['Name'].value_counts().index.map(lambda x: ind_col_map[name_ind_map[x]])

Finally, you only need to simply add color to your plotting function:

ax = df['Name'].value_counts().plot(kind='bar',
                                    figsize=(14,8),
                                    title="Number for each Owner Name", color=c)
ax.set_xlabel("Owner Names")
ax.set_ylabel("Frequency")
plt.show()