No, it doesn't. The Smiths never had the widespread acceptance the Ramones had nor is there a sub-genre of music they are responsible for like TheRamones were.

You have a FUCKLOAD of bands from all over the world that do Ramones inspired punk. There are almost no bands trying to be the Smiths and given what a sack of shit Morrissey has become I doubt he will be remembered for very long once he stops performing.