Equal risk weighting replaces "same dollars each" with "same risk each". In discretionary trading that usually means the same dollar-risk to the stop; in systematic portfolios it means the same volatility contribution.
Worked example on $100,000 with a 0.5% target per name. Stock A moves $2 a day, so you hold 250 shares ($500 / $2). Stock B moves $0.25 a day, so you hold 2,000 shares. A is a $12,000 position and B is a $30,000 position, yet both swing about $500 on a normal day.
This is a better starting point than equal-dollar-weighting, but it is still naive: it ignores correlation. Five equally risk-weighted positions in the same sector are one position wearing five hats, which is what equal-risk-contribution and risk-parity try to fix.
Related: equal-dollar-weighting, equal-risk-contribution, risk-parity, correlation