The shower may not be constructed per codes. It is wrong, but not uncommon to put the waterproof layer flat on the floor, then build up the pan for tile. By code, the waterproof layer must be installed on a slope (and the tile ISN'T the waterproof layer). If yours is constructed that way, over the years, the pan will retain moisture, and it is really hard to get rid of. Mold needs moisture and food. killing the mold then sealing the grout might help, but grout sealant isn't waterproof, either.
IMHO, a new shower built with a surface membrane, such as Kerdi from
www.schluter.com puts a waterproof layer right under the tile all over, including the pan. Therefore, there's little underneath that can absorb moisture (i.e., no deckmud on the floor or cbu in the walls), so it dries out much quicker and is easier to maintain. If you tile with a good porcelain tile (which has typically much less than a 1% absorbtion rate), it's pretty bulletproof. Then, if you use an epoxy grout and the corner details (expansion joints) from schluter, there'd be NO caulk in the shower to get funky, either.
Try wiping the shower down after use, and running the vent fan during and after the shower to help dry things out. No moisture, no mold. A mix of water and bleach will help kill it. Then, you must keep extraneous soap scum and crud off, so a good cleaning doesn't hurt periodically. That kills the two things easiest to manage - food and moisture.