Regardless of single- or multi-curve framework, you can always think of a vanilla, fixed-to-float interest rate swap as a linear combination of a long (short) fixed rate bond and a short (long) floating rate note. The floating rate note has an overall DV01 of close to zero since the coupons adjust periodically alongside the discount factors. Since you don't have this offsetting effect on the fixed rate bond, most of the risk/duration/DV01 is concentrated on the redemption date as the principal is paid back. Depending on how you build your curve and the swap is set up, the key rate risk will not be exactly zero on each tenor, but close to it.
If rates are close to zero the DV01 scales roughly with a factor of 1`000 on 10MM notional: for example, 10y swap ~ 10k DV01, 20y swap ~ 20k DV01 etc. Again this is more of heuristic.
A formula and proof for DV01 can be found here.