create legend for markercolor and size

The key is to capture the artist (PathCollection in this case) returned by the scatter command. That has a method to retrieve the legend items manually, and it has keywords to distinguish between size and color (default). The num keyword can be used to reduce the amount of items returned, which is useful in the case of a (semi)continuous property as the size can be.

The example below plots two separate legends for both properties. You can also combine the handles and labels of both and plot them in a single legend if needed.

import matplotlib.pyplot as plt
from matplotlib.patches import Rectangle
import cartopy.crs as ccrs
import cartopy
import numpy as np

lons = np.random.randint(-170, 170, 100)
lats = np.random.randint(-80, 80, 100)
sizes = np.random.rand(100) * 100 + 5
colors = np.random.randint(0, 3, 100)

fig, ax = plt.subplots(
    figsize=(8,4), dpi=86, facecolor='w', 
    subplot_kw=dict(projection=ccrs.PlateCarree()),
)
fig.subplots_adjust(left=0, right=1, bottom=0, top=1)

ax.add_feature(cartopy.feature.LAND, ec='none', fc='#EEEFEE', label='country border')
ax.add_feature(cartopy.feature.BORDERS, ec='w', fc='k', lw=0.75, label='country border')
ax.coastlines(resolution='10m', color='#EEEFEE', lw=0.75)

m = ax.scatter(lons, lats, s=sizes, c=colors, zorder=5, label="points")

l1 = ax.legend(
    *m.legend_elements(prop="colors", num="auto"), title="Colors", framealpha=1, 
    loc="upper right", bbox_to_anchor=(0.88, 0.8, 0.12, 0.2), mode="expand",
)
ax.add_artist(l1) # prevent overwriting with second legend

l2 = ax.legend(
    *m.legend_elements(prop="sizes", num=5), title="Sizes", framealpha=1,
    loc="upper right", bbox_to_anchor=(0.88, 0.55, 0.12, 0.2), mode="expand",
)

The documentation about this shows some variations on this:
https://matplotlib.org/stable/gallery/lines_bars_and_markers/scatter_with_legend.html#automated-legend-creation

The answer by Rutger Kassies is excellent for many use cases. However, he mentions that One can also combine the handles and labels of both and plot them in a single legend if needed.

Here I offer another answer that shows the steps to create the single legend manually. Inside the single legend, 2 groups of sub legends are created and arranged as needed.

With single legend, you don't need to find the values of bbox_to_anchor for the second (or third and so on) to position them properly.

With manual creation of items into a single legend, you have full control of the items' you need in the legend. However, it need some extra coding to achieve the goal.

import matplotlib.pyplot as plt
import matplotlib.lines as mlines
import cartopy.crs as ccrs
import cartopy.feature as cfeature

# For `Categories` symbol
# Each item of legends requires 3 properties: color/text/marker_shape
color_V = ["green", "orange", "purple", "red", "cyan", "magenta"]
text_V = ["cat_4", "cat_9", "cat_13", "cat_15", "cat_19", "cat_33"]
marker_V = ["o", "o", "o", "o", "o", "o"]
len_V = len(color_V)

# For `Size/values` symbol
color_S = ["gray", "gray", "gray", "gray"]
sizes_S = [4, 8, 12, 16]         #increasing values ...
text_S = ["4", "8", "12", "16"]  #cover `sizes1` below
marker_S = ["o", "o", "o", "o"]  #use disk shape
len_S = len(color_S)

# Demo data locations and attributes
xs = [23,12,4,25,24,52,17,33]
ys = [41,12,32,15,35,21,23,43]
colors1 = ["green", "orange", "purple", "red", "cyan", "magenta", "green", "orange"]
#texts1 = ["4", "9", "13", "15", "19", "33", "4", "9"]
markers1 = ["o", "o", "o", "o", "o", "o", "o", "o"]
sizes1 = [10,16,9,12,7,4,2,6]
len1 = len(xs)

all_patches = [] #for items in a single legend

# Create figure and `ax` for map plotting
# This form can create a single axes or an array of axes
fig, ax = plt.subplots(nrows=1, ncols=1, figsize=(8,6), subplot_kw={'projection': ccrs.PlateCarree()})

# All steps of plots will be done on `ax`

# [1] Add an invisible object as a spacer in the legend box
#rect = mpatches.Rectangle([0, 0], 0.01, 0.01, ec="none", color="lightgray")
all_patches.append(mlines.Line2D([0, 0], [1, 0], color="none"))

# Explicitly defining the elements in the legend
# [2] Add proxied text: 'Categories' to the legend
line = mlines.Line2D([0, 0], [1, 0], lw=.5, alpha=0.9, color="none")
line.set_label('Categories')  # Title for 1st group of symbols in the legend
all_patches.append(line)

# [3] Plot (on the axes) `none` data point and 
#  save the output patches for `Categories` group
patches_V = [ ax.plot([],[], marker=marker_V[i], ms=8, ls="", color=color_V[i], \
          label="{:s}".format(text_V[i]) )[0] \
          for i in range(len_V) ]
all_patches += patches_V

# [4] Add an invisible object as a spacer in the legend box
all_patches.append(mlines.Line2D([0, 0], [1, 0], color="none"))

# [5] Add proxied text: 'Sizes' to the legend
x, y = ([0, 1], [0, 0])
line = mlines.Line2D([0, 0], [1, 0], lw=.5, alpha=0.9, color="none")
line.set_label('Sizes')  # Title for 2nd group of symbols in the legend
all_patches.append(line)


# [6] Create patches for `Sizes` group
patches_S = [ ax.plot([],[], marker=marker_S[i], ms=sizes_S[i], ls="", \
        color=color_S[i], \
        label="{:s}".format(text_S[i]) )[0] for i in range(len_S) ]

all_patches += patches_S

# Plot point data using the demo data
for i in range(len1):
    ax.plot(xs[i], ys[i], marker=markers1[i], ms=sizes1[i], color=colors1[i])

ax.set_extent([0, 80, 0, 60])

# Plot the legend in the upper-right corner
combined_legend = ax.legend(handles=all_patches, 
                            bbox_to_anchor=(1, 1), 
                            title="The Legend", 
                            loc='upper right', 
                            ncol=1, 
                            numpoints=1, 
                            facecolor="lightgray", 
                            fontsize = 10, 
                            title_fontsize= 12, 
                            labelspacing = 0.55, 
                            shadow=True)

# Draw some basemap features
ax.coastlines(lw=0.3, color="k")
ax.add_feature(cfeature.LAND)
ax.add_feature(cfeature.OCEAN)
plt.title("Legend for Categories and Sizes")
plt.show()

The output map: