If a sorcerer multiclasses into a prestige class that furthers their spellcasting progression, like the lore master for example, do they continue to earn their bonus bloodline spells?
I would imagine so.
But doesn't it count as a class ability separate from spellcasting?
It's a part of the bloodline ability, which isn't advanced by PrC's (except the Dragon Disciple and that new class that gives you everything past second level) since they advance Spellcasting Progression only.
Note that the dragon disciple is the only prestige class that says it grants (additional) "bonus spells" from a bloodline. All other prestige classes state that the character does not "gain other benefits a character of that class would have gained, except for additional spells per day, spells known (if she is a spontaneous spellcaster), and an increased effective level of spellcasting."
The sorcerer class really should say something similar to the oracle. It doesn't make much sense for a bloodline to stop progressing.
I don't think they do.
If a sorcerer multiclasses into a prestige class that furthers their spellcasting progression, like the lore master for example, do they continue to earn their bonus bloodline spells?
No, you only advance in terms of spellcasting, e.g. your spell and caster level progresses normally, but the bonus spells from bloodlines increase with sorcerer levels as opposed to spell or caster levels.