[BUG] issues with dropdown persistence in dash 3.0

[revmlx]

Describe your context
I have recently updated my dash app to dash 3.0 and noticed the persistence on my dropdowns was gone.

I have a virtual environment (python 3.10) with dash 3.

dash                      3.0.1
dash_ag_grid              31.3.1
dash-bootstrap-components 2.0.0
dash-chart-editor         0.0.1a5
dash-core-components      2.0.0
dash-defer-js-import      0.0.2
dash-extensions           1.0.20
dash-html-components      2.0.0
dash_mantine_components   1.1.0
dash-table                5.0.0
dash_uploader             0.7.0a2

I also have an older environment (python 3.11) with an older version of dash:

dash                      2.14.1
dash-ag-grid              2.4.0
dash-bootstrap-components 1.5.0
dash-core-components      2.0.0
dash-defer-js-import      0.0.2
dash-extensions           1.0.4
dash-html-components      2.0.0
dash-mantine-components   0.12.1
dash-table                5.0.0
dash-uploader             0.6.0        

Describe the bug
Initially I figured it had to do with the way I had my callbacks setup (we have many and quite complicated ones) so I grabbed the example from the site and ran it standalone:

from dash import Dash, dcc, html, Input, Output, callback

CITIES = ['Boston', 'London', 'Montreal']
NEIGHBORHOODS = {
    'Boston': ['Back Bay', 'Fenway', 'Jamaica Plain'],
    'London': ['Canary Wharf', 'Hackney', 'Kensington'],
    'Montreal': ['Le Plateau', 'Mile End', 'Rosemont']
}

app = Dash()

app.layout = html.Div([
    'Choose a city:',
    dcc.Dropdown(CITIES, 'Montreal', id='persisted-city', persistence=True),
    html.Br(),

    'correlated persistence - choose a neighborhood:',
    html.Div(dcc.Dropdown(id='neighborhood'), id='neighborhood-container'),
    html.Br(),
    html.Div(id='persisted-choices')
])


@callback(
    Output('neighborhood-container', 'children'),
    Input('persisted-city', 'value')
)
def set_neighborhood(city):
    neighborhoods = NEIGHBORHOODS[city]
    return dcc.Dropdown(neighborhoods, neighborhoods[0], id='neighborhood',
        persistence_type='session',
        persistence=city
    )


@callback(
    Output('persisted-choices', 'children'),
    Input('persisted-city', 'value'), Input('neighborhood', 'value')
)
def set_out(city, neighborhood):
    return f'You chose: {neighborhood}, {city}'


if __name__ == '__main__':
    app.run(debug=True, port=8080, host="0.0.0.0")

and persistence didn't hold.
I then tested it with the old venv (dash 2) and there it does work. Even more peculiar, if I then switch to the new venv again (dash 3), it remembers the persistent values from what I changed with dash 2. But it will not remember anything I change with dash 3.

Expected behavior

I expected to have persitsence work as it is explained in the tutorial on the website for both dash 2 and dash 3

[BSd3v]

Hey @revmlx,

I believe that my PR may address this: #3248