Finite can be in contrast with infinite, but also in contrast with a point, zero length,as John pointed out. As the standard model has point particles by postulate, making the particles of one dimension of finite length is in contrast to the zero length of the standard model of particle physics
The statement : "Thus the string is the only truly 'fundamental' particle" is a mathematical generic statement, not a physics one. In the mathematics of string theory the standard model can be embedded, elementary particles have a one to one correspondence, with particular vibrational energy levels of the string ( we are in quantum mechanics after all, or should be) .
The physics of the standard model does not stop at the mathematical description but also interaction and conservation of quantum states rules have to hold. One cannot turn a string representing an electron to represent a proton, unless strict interactions take place conserving lepton and quark numbers, by transferring them to other strings.