It can be a confusing scene, but early pieces of dialogue easily hint towards the answer. Why did Walter White make his son drink?
Walter White makes his son drink due to a sense of insecurity. His cancer remission is leading towards a “normal” life again, so he allows his son to drink to feel in control and powerful.
Hank, who tries to stop Walter, signifies the authority that Walter has tried to escape. Showing that he’s the “man of the house” by letting his son drink is a desperate attempt to regain the control that being a criminal allowed him to have.
The Whites celeberate Walter’s remission
In Breaking Bad: Season 2, Episode 10, the Whites host a family party to celebrate Walter’s cancer remission. Although it’s meant to commemorate his good health, Walter is secretly depressed by his diagnosis.
The reason that Walter manufactures narcotics and wraps himself up in a criminal life is because he thought he’d soon pass away. Now, realizing that he’ll live, Walter feels anger that his blaze-of-glory plan is kaput.
Walter secretly enjoyed the freedom of the criminal life. Now, remission free and soon to return to teaching, Walter is having a difficult time coping with the horrible things. He was under the assumption that he wouldn’t be around to face the consequences.
The party goes fine at first, but Walter lightly admits in a speech that he thought “why me” when the remission began. Later, Walter, Walt Jr., and Hank are drinking hard liquor in the backyard.
Walter allows his son to drink, Hank disagrees
At first, Hank is OK with Walt Jr. having a celebratory shot of liquor with his family. Things take a turn for the worst, however, when Walter insists that his son be allowed to keep drinking.
Hank, concerned for Walt Jr.’s young age and lack of drinking experience, asks Walter to stop. Walter keeps pouring drinks, so Hank takes the bottle and walks away.
In a fit of anger, Walter says it is his house, his son, and his bottle. It is true that Walter is angry over his remission, but he also feels a sense of sensitive insecurity that another man is telling him how to raise his son.
Walter taking control over his life was one of the most appealing traits of turning to the criminal life. Now that his life is back to “normal” and he’s being told how to live again, he snaps out of anger.
So that’s the answer. Walter White makes his son drink in order to feel in control and powerful again. His remission is leading Walter back to a “normal” life where he has no power.