Does the number of links of a chainsaw chain affect the sprocket?

Presumably the new bar is either longer or wider or both than the old bar? Number of links is to fit the bar - pitch is to fit the sprocket.

Gauge is the thickness of the drive tangs on the saw chain. Chain gauge must match the groove width on the bar.

The only effect chain gauge will have on standard spur sprockets is the contact area it presents on the spur teeth, thicker gauge = wider contact area.

Rim Sprockets use a pocket that must be wide enough to allow the drive tangs to seat.

Pitch is the length of the links that make up the chain, basically the distance spanned by three rivets divided by 2 and chain pitch must match the sprocket pitch so the chain is properly driven.

The reason for measuring over the span of three rivets and taking the average to calculate pitch is that the drive link has the rivets closer together than the tooth and span links.

Where the pitch matters, as shown on the simpler spur sprocket chain drive mechanism on an electric pole saw. Chain drive tangs must be the right distance apart to engage the sprocket teeth properly.

This leaves the remaining variable which is the number of links. More links, longer chain which must of course be accommodated by having a longer bar. Since the chain is seven links longer (81-74), the chain will be about 2.3" longer. If the bar nose radius on the new bar is equal to that of the old bar, your new bar should be about 1" longer.

Also, there are different geometries on saw bars that can be used to take up that extra length which only can be seen by comparing the old bar against the new bar.

Since the number of links is larger(81 instead of 74), there is low chance that it would fit properly.

What I suggest is to check the chain details(gauge,pitch and drive link) before you get one. You can find the details printed on the bar or in the manual.

If not, take the chain with you to a local hardware store, they can tell you all the details just taking a look at the chain.