Setting equal margins while keeping range of ticks as defined

You can use plt.ylim(ymin, ymax) and plt.xlim(xmin, xmax) to set the same plot size. Here is a code that works:

# Get the max value in both y axis. You could also to this for the x axis
# as well as get the minimun value. Here I'm just assuming it's zero
y1max = max(y1)
y2max = max(y2)
ymax = max(y1max, y2max)

plt.figure()
plt.scatter(x = x1, y = y1, marker = '.', s = 1.2, alpha = 0.75, label = 'DATA', color = 'darkorange')
plt.yticks(np.arange(0,1.2,0.2))
plt.xticks(np.arange(0,420,60))

# After you create the figure, set the y max and min values
plt.ylim(0, ymax)

lgnd = plt.legend(loc=1, scatterpoints=1, fontsize=10)
lgnd.legendHandles[0]._sizes = [30]
plt.show()

plt.figure()
plt.scatter(x = x2, y = y2, marker = '.', s = 1.2, alpha = 0.75, label = 'DATA')
plt.xticks(np.arange(0,420,60))
plt.yticks(np.arange(0,1.2,0.2))

# Do the same for the second plot, with the same ymax value
plt.ylim(0, ymax)

lgnd = plt.legend(loc=1, scatterpoints=1, fontsize=10)
lgnd.legendHandles[0]._sizes = [30]
plt.show()

Here is what i got. My data is not in the same range as yours, so the ticks don't go until the end, but the y axis scale is the same, as you can see.

If I understand your question correctly, you are looking for plt.ylim and plt.xlim, which allows to setup the minimum and maximum of axis. Additionally, you can include figsize option within plt.figure() to ensure that both of the generated figures are of equal sizes. Here is an example based on your code:

import matplotlib.pyplot as plt
import numpy as np

#generating test data
x1 = np.random.random((300,1))
x2 = np.random.random((300,1))
y1 = np.random.random((300,1))
y2 = np.random.random((300,1))

#setting the size of the figure
plt.figure(figsize=(8,6))
plt.scatter(x = x1, y = y1, marker = 'o', s = 3.2, alpha = 0.75, label = 'DATA', color = 'darkorange')
#plt.yticks(np.arange(0,1.2,0.2))
#plt.xticks(np.arange(0,420,60))
lgnd = plt.legend(loc=1, scatterpoints=1, fontsize=10)
lgnd.legendHandles[0]._sizes = [30]
# plt.margins(y = 0.02)

#setting minimum and maximum of the plots
plt.ylim([0,1])
plt.xlim([0,1])
plt.show()

#setting the size of the figure
plt.figure(figsize=(8,6))
plt.scatter(x = x2, y = y2, marker = 'o', s = 3.2, alpha = 0.75, label = 'DATA')
#plt.xticks(np.arange(0,420,60))
#plt.yticks(np.arange(0,1.2,0.2))

#setting minimum and maximum of the plots
plt.ylim([0,1])
plt.xlim([0,1])
lgnd = plt.legend(loc=1, scatterpoints=1, fontsize=10)
lgnd.legendHandles[0]._sizes = [30]
# plt.margins(y = 0.02)
plt.show() 

Output:

Side by Side: