Hi nadenislamarre I'm using 5.18 X86_64 version of Batocera and I feel there's some problem with hiscore saving with Mame 2016 core. You may be interested in fixing it in 5.19 release.
The problem in short:
1) I copied hiscore.dat file (the correct one, for Mame 2016 aka 0.174) in the path share > bios > mame2016
2) edited the retroarch-core-options.cfg (path: system > configs > retroarch > cores) adding the following lines
mame2016-external_hiscore = "enabled"
mame2016_alternate_renderer = "disabled"
mame2016_auto_save = "disabled"
mame2016_boot_from_cli = "disabled"
mame2016_boot_to_bios = "disabled"
mame2016_boot_to_osd = "disabled"
mame2016_cheats_enable = "disabled"
mame2016_media_type = "rom"
mame2016_mouse_enable = "disabled"
mame2016_read_config = "disabled"
mame2016_saves = "game"
mame2016_skip_gameinfo = "enabled"
mame2016_skip_nagscreen = "enabled"
mame2016_skip_warnings = "enabled"
mame2016_softlists_auto_media = "enabled"
mame2016_softlists_enable = "enabled"
mame2016_throttle = "disabled"
mame2016_write_config = "disabled"
But, as you guessed, my hiscores aren't saved yet. I noticed that the path "share > roms > mame" has a "mame 2003" folder with subfolder "hi" for storing hiscores. There isn't such structure per mame 2016 core, it could be the source of the problem.
If you and the developers' team have time, I'd appreciate if you could investigate that in order to solve. I'll be ready to test. Thanks a lot