Small businesses often have slightly dysfunctional management or internal systems that don't make much sense to someone new but are "how they've always done it". If you're going to work for one, you have to accept that there will be differences to working for a big business where everything is standardised.
I think it sounds about 50/50 fault- ideally the manager would have checked you received your shifts, but equally, you could have been more proactive. It's possible she took you asking for time off as indication that you'd seen your shifts. Just as an FYI it's generally not ideal to be asking for time off during your first couple of months of employment.
I would personally chalk this up to a miscommunication, apologize and move on. If you are lucky, the manager will apologize back, but don't count on it happening.
I'd also make the effort to appear very reliable over the next few months.