[RPG] Do multiple Speed weapons grant multiple extra attacks

No, I could not find a bonus to multiple limbs in the rules.

If you're looking for a house rule, you could say you get a +2 to CMB for every additional limb beyond 2 (like animals get to CMD for being on all 4's).

However, if this is an eidolon or a creature you're defining, adding the Grab ability and MultiAttack monster feat would be hugely beneficial:

Grab (Ex) If a creature with this special attack hits with the indicated attack (usually a claw or bite attack), it deals normal damage and attempts to start a grapple as a free action without provoking an attack of opportunity. Unless otherwise noted, grab can only be used against targets of a size equal to or smaller than the creature with this ability. If the creature can use grab on creatures of other sizes, it is noted in the creature's Special Attacks line. The creature has the option to conduct the grapple normally, or simply use the part of its body it used in the grab to hold the opponent. If it chooses to do the latter, it takes a –20 penalty on its CMB check to make and maintain the grapple, but does not gain the grappled condition itself. A successful hold does not deal any extra damage unless the creature also has the constrict special attack. If the creature does not constrict, each successful grapple check it makes during successive rounds automatically deals the damage indicated for the attack that established the hold. Otherwise, it deals constriction damage as well (the amount is given in the creature's descriptive text).

Creatures with the grab special attack receive a +4 bonus on combat maneuver checks made to start and maintain a grapple.

Format: grab; Location: individual attacks.

Emphasis mine.

Source: Paizo

Yes, they stack.

Bestial Aspect says:

If the rageshaper's bloodrage powers already grant natural attacks or alternate modes of movement, then the bonuses granted by bestial aspect also apply to these bloodrage powers.

So, its bonuses apply to the claws granted by Draconic Bloodline.

While the damage progression of the claws is similar to an effective size increase, it is also similar to the damage progression of a monk's Unarmed Strike.
(Which could maybe be argued as an effective size increase too, but that's irrelevant because...)

Bestial Aspect however, doesn't bear any resemblance to a size increase. Your die doesn't "increase by one step", it increases by one die. Meaning, your 1d6 claws become 2d6; your 1d8 claws become 2d8. So there is no stacking of effective size increases.