The fact things were on a phone line helped in some ways (exclusivity, time, identifiability), but doing it all through a mobile phone or other telephony does not really capture that. For one, there would only be single node BBSs unless you got creative, so I think the mobile phone part is too literal on the concept of phones.
Secondly, while this is less popular, a lot of us including myself were not actually identifiable unless we wanted to be. This happened because you could call a BBS through an intermediary such as a PBX, and it was common to war dial to find and hack these (or use open hardware) to dial out long distance. Another way was carding (stealing credit cards) to pay for calls, which would also often answer caller ID requests with another number than your own depending on what you bought.
The anonymity was more complicated as you can see, but yes, it did help quite a bit to have some sort of reputation attached to your name. A BBS handle could carry weight across many systems, and word did spread. Some Sysops were pretty nasty to people and went out of their way with that, but I think most people accepted it and didn't do things like try to take them to court like we have today. Somehow, whether they knew who you were or not, it just was not worth it and getting other users to accept you in one form or another was important. This happened via message bases, uploads (try to be top, avoid being banned for bad uploads), new user signup, games, and more. So souring your reputation had a cost if you were an avid BBSer, and even more so if you had met people in real life.
I think anything remotely similar to BBSs in terms of success needs to do at least some of these things:
- No frills access
- Tie things together as a cohesive hole
- "Character" per system. Logging on to X should not be the same as logging on to Y, but familiar enough to still be usable.
- Distributed (run by a person/people)
- No advertising. Advertising is at the root of ruining a lot
- Cross-over between the real-world and the virtual-world. Ex: In person meetings.
- Sense of urgency
- Connections to several rabid groups of sub-culture that will drive future updates, modding, usage, etc. In the old days, this was HAM Radio operators, ANSI artists, Anarchists, Punks, Crackers, Warez kids, Phreakers, D&D players, and many other groups.
- Decent give and take type systems. Ratios, time, and chat often worked this way and sometimes were all interconnected. The more you upload, the more time you get, and the more you can chat for example.
Many more things, but I'll stop there.