Curves intended for both signing and key agreement
Technically, any elliptic curve can support both signing and key agreement. However, Curve25519 as described in the original paper is meant for $x$-only arithmetic. This means that one never computes the $y$-coordinate. The issue is that you need the $y$-coordinate for signature verification in the standard algorithms (ECDSA, EdDSA).

Therefore, if you want to do signatures with the Curve25519 curve, you need to compute the $y$-coordinate. There are different ways to do this, one is to simply leave the curve as is (in "Montgomery form") and compute its $y$-coordinate. A different way, is to transform the curve into a new shape (the "twisted Edwards form") and do arithmetic there. This new curve is called Ed25519, and it is "(birationally) equivalent" to the old curve. There are even more flavours.

Again, the choice you make will be for efficiency and security. Not because one curve can be used for signatures while the other can't. I suppose the $y$-coordinate computation on Curve25519 is slow enough for people to prefer using Ed25519 instead.