It sounds like you have the old cup-and-cone type of bottom bracket. Here are a couple of links there are tons more online:
https://www.sheldonbrown.com/tooltips/bbcups.html
https://www.sheldonbrown.com/tooltips/bbadj.html
You can either service the existing BB or remove it and replace it with a cartridge type, which has the splines you were expecting. There are tips on removing the old one in the linked article, but it might be most efficient to have a shop do the work (or go to a co-op) rather than buy tools that you will only use once, if you decide to install the cartridge type. I have successfully removed the cup and cone type with a large adjustable wrench for the drive side and a hammer and punch for the non-drive side but it can be difficult work with improvised tools.