A lot of companies do, some in some balls but not in others.  Some Lane 1 balls are available in 13#, some aren't.  Many Brunswick balls are available in 13# with the same (or weight-appropriate) core as the heavier weights.

You can usually tell if the website or spec sheet lists the diff numbers for the different weights, as there's often a big jump when they switch cores.

SH

These have the original cores in 13lb balls for sure:

Brunswick Inferno line
Brunswick BVP line
Storm Master line

I'm not sure for the others.

However, it's NOT the SHAPE of the core that's most important.

It's the RG and differential it creates.

You can still have the original core in a 13lb Smoking Inferno but the ball's RG must have gone way higher (because the cover may already have contributed to over 10lb of the weight). The mass bias may not be as strong as you see in a 14lb one.

So, when choosing 13lb equipments be sure you take a look at the actual specification of the ball at that weight.