One possibility would be to NOT use lmplot(), but directly use regplot() instead. regplot() plots on the axes you pass as an argument with ax=.
You lose the ability to automatically split your dataset according to a certain variable, but if you know beforehand the plots you want to generate, it shouldn’t be a problem.
Something like this:
import matplotlib.pyplot as plt
import seaborn as sns
fig, axs = plt.subplots(ncols=3)
sns.regplot(x='value', y='wage', data=df_melt, ax=axs[0])
sns.regplot(x='value', y='wage', data=df_melt, ax=axs[1])
sns.boxplot(x='education',y='wage', data=df_melt, ax=axs[2])