You can create a custom discrete colorbar quite easily by using a BoundaryNorm as normalizer for your scatter. The quirky bit (in my method) is making 0 showup as grey.

For images i often use the cmap.set_bad() and convert my data to a numpy masked array. That would be much easier to make 0 grey, but i couldnt get this to work with the scatter or the custom cmap.

As an alternative you can make your own cmap from scratch, or read-out an existing one and override just some specific entries.

import numpy as np
import matplotlib as mpl
import matplotlib.pylab as plt

fig, ax = plt.subplots(1, 1, figsize=(6, 6))  # setup the plot

x = np.random.rand(20)  # define the data
y = np.random.rand(20)  # define the data
tag = np.random.randint(0, 20, 20)
tag[10:12] = 0  # make sure there are some 0 values to show up as grey

cmap = plt.cm.jet  # define the colormap
# extract all colors from the .jet map
cmaplist = [cmap(i) for i in range(cmap.N)]
# force the first color entry to be grey
cmaplist[0] = (.5, .5, .5, 1.0)

# create the new map
cmap = mpl.colors.LinearSegmentedColormap.from_list(
    'Custom cmap', cmaplist, cmap.N)

# define the bins and normalize
bounds = np.linspace(0, 20, 21)
norm = mpl.colors.BoundaryNorm(bounds, cmap.N)

# make the scatter
scat = ax.scatter(x, y, c=tag, s=np.random.randint(100, 500, 20),
                  cmap=cmap, norm=norm)

# create a second axes for the colorbar
ax2 = fig.add_axes([0.95, 0.1, 0.03, 0.8])
cb = plt.colorbar.ColorbarBase(ax2, cmap=cmap, norm=norm,
    spacing='proportional', ticks=bounds, boundaries=bounds, format='%1i')

ax.set_title('Well defined discrete colors')
ax2.set_ylabel('Very custom cbar [-]', size=12)

I personally think that with 20 different colors its a bit hard to read the specific value, but thats up to you of course.

A simple option to replace matplotlib.cm.get_cmap (deprecated in 3.7.0) is to use matplotlib.pyplot.get_cmap (that was preserved in the API for backward compatibility).

Another one woud be to make a resampled colormap (used under the hood) :

resampled(lutsize) [source]

Return a new colormap with lutsize entries.

Note that the lut is a parameter in get_cmap(name=None, lut=None).

    cm = mpl.cm.get_cmap("viridis", 7)   # (-)
    cm = plt.get_cmap("viridis", 7)      # (+)