Probably because the Overton window leans right enough that even moderate leftwing policies feel leftist while moderate rightwing policies feel moderate, making center-right a useful label in one case and not in the other. Or it’s a bourgeois capitalist conspiracy, probably both.

The parties have different identities and positions. However they always change names, merge and split. There is also a long tradition of forming alliances of different parties. For example Emmanuel Grégoire who is currently leading in Paris is a Socialist, but is also supported by the Communists and some other smaller parties.

Probably an agenda, it’d make more sense to say centre-left, left.