If a Cloth Armor is attached to a Brutalizer and that Brutalizer gets attached to a unit, does the unit get the shield 2 from the Cloth Armor?

No — you can't build that stack at all, and even hypothetically the unit wouldn't get the Shield 2.

Equip and Quick-Draw both attach a gear to a unit you control (Rule 818.1.c.2), so there is no legal way to put Cloth Armor onto Brutalizer — another piece of gear can't be the target of an Equip ability, and no effect in the game attaches equipment to equipment.

For the hypothetical anyway: an attached card appends its Effect Text to the Top-Most card it is attached to (Rule 718.3), so Cloth Armor's Shield 2 would be appended to Brutalizer's rules text, not to whatever Brutalizer later attaches to. The host unit would only pick up Brutalizer's own effect text — +2 [M] if it was attached to that unit this turn.

In normal play you just attach both to the unit: the unit has Shield 2 while defending, plus Brutalizer's +2 [M], and they stack fine.