In newer Seaborn versions (≥0.11) you can use:

colors = ["#090364", "#091e75", "#093885", "#085396", "#086da6", 
           "#0888b7", "#08a2c7", "#07bdd8", "#07d7e8", "#07f2f9", 
           "#f9ac07", "#c77406", "#963b04", "#640303"]

sns.set_style("whitegrid")

plot = sns.scatterplot(
    x='length', 
    y='type',
    hue='temperature',
    legend=False, 
    palette=sns.color_palette(colors),
    s=200,
    data=df)

The reason why your custom palette isn’t picked by Seaborn is that sns.set_palette() only sets the default color cycle for plots that don't use the hue semantic.

When you use hue=… in Seaborn, data points are categorised using different colors based on the hue variable and its categories (or levels). For instance, if hue='temperature', Seaborn extracts all unique distinct values in the 'temperature' column.

Here’s the thing: you need to specify a color for each of these hue levels and it’s up to you to make sure that your custom palette has the correct number of colors.

From your code I can also tell that you’re using an older version of Seaborn, certainly pre because starting from version 0.11:

Most plotting functions now require all of their parameters to be specified using keyword arguments.

(quote from https://seaborn.pydata.org/whatsnew/v0.11.0.html#required-keyword-arguments)

So the positional arguments

plot = sns.scatterplot(
    df.loc[:,'length'], 
    df.loc[:,'type'],

in newer Seaborn versions should be given as

plot = sns.scatterplot(
    x=df.loc[:,'length'], 
    y=df.loc[:,'type'],

Here's a demo with an extra improvement of the original code (using data=df so that the columns can be specified just by their name):

import seaborn as sns
import matplotlib.pyplot as plt
import pandas as pd
import numpy as np

# Create a simple sample DataFrame
np.random.seed(0)
temperature_levels = [f'Temp{i:02d}' for i in range(14)]  # 14 unique temps

df = pd.DataFrame({
    'length': np.random.rand(100) * 10,
    'type': np.random.rand(100) * 5,  
    'temperature': np.random.choice(temperature_levels, size=100)
})

# 14 colors
colors = ["#090364", "#091e75", "#093885", "#085396", "#086da6", 
           "#0888b7", "#08a2c7", "#07bdd8", "#07d7e8", "#07f2f9", 
           "#f9ac07", "#c77406", "#963b04", "#640303"]

sns.set_style("whitegrid")

plot = sns.scatterplot(
    x='length', 
    y='type',
    hue='temperature',
    legend=False, 
    palette=sns.color_palette(colors),
    s=200,
    data=df)

What happens if the number of colors doesn't match the number of hue levels?

UserWarning: The palette list has more values (14) than needed (10), which may not be intended.

If the number of hue levels is larger than the number of colors in your palette, Seaborn will cicle on the colors thus creating unwanted effects. You may also see a warning like this:

The palette list has fewer values (14) than needed (43) and will cycle, which may produce an uninterpretable plot.