I do think Dany should have been told this at some point. Even if only tangentially.
However, there is some justification that I can find in the TV show. I'm omitting the books as I haven't read them.
Danaerys' followers are comprised of many different groups. Westerosi, Dothraki, Unsullied...
However, their followers all share a common theme: They were rejected by their "home" and were in need of a second chance. Dany gives these people a second chance.
Jorah and Selmy are the clearest examples of this, but it applies to most involved parties. The one exception I can think of is Daario, who follows Dany out of love, not necessity.
There were Dothraki who refused to follow Daenerys after the death of Khal Drogo. Logically speaking, these will be Dothraki that still swear by traditional Dothraki values; e.g. not considering a woman as an equal, let alone a leader.
This means that the Dothraki who stay with Daenarys choose to forego Dothraki tradition, and choose Daenerys instead. Everyone could have their own reason, but I think the general recurring incentive for the remaining Dothraki is that they need a home, which Dany is willing to provide.
Informing Dany that she should become a Dosh Khaleen goes directly against that idea. This would make Dany unavailable as a leader, and therefore unable to (personally) provide the refuge that most of her followers are seeking.
So there are many possible reasons why Dany did not join (nor know of) the Dosh Khaleen, until the topic was raised by Khal Moro. I'll try to list them as understandably as possible:
Note: The reasons listed below are possibilities. Not facts. Just making sure...
If anyone has corrections that exclude possibilities (based on the TV show), please do comment!
1. She was never told about this tradition.
As a general comment, I think this is unlikely, but not impossible. Dany is young and naive, and she has her hands full with ruling her people. Her advisors may have shielded her from this.
- Her followers do not want to lose their leader, and have no reason to suggest she joins the Dosh Khaleen.
- No Dothraki (other than Dany's followers) would respect Daenerys' rule anyway, so no one was hoping to visit Vaes Dothrak anytime soon. There was no reason to comply with Dothraki customs if they avoid the Dothraki altogether.
- Her followers did not think that the same rules would apply to Daenerys, a Westerosi. She was already an exception to most Dothraki customs (e.g. ruling as Khaleesi alone), so why not this one?
- Daenerys' general pro-feminist attitude is interpreted by the Dothraki as a complete rejection of Dothraki (gender) tradition; and no one expected her to comply with these traditions anyway.
2. She was told that this is Dothraki tradition
This seems much more likely to me. Even if she was never directly told that she is expected to join the Dosh Khaleen, she probably would have heard about the custom at one point or another.
If she hasn't even heard of her own people's traditions, that shows a lack of interest in their culture. Which doesn't seem like the Daenerys that we see on the show (although she is woefully naive at times, so it's not impossible).
- Daenerys does not think woman are lesser to men. Following that idea, why should she give up her throne and follow traditions that she thinks are outdated? She has already chosen to ignore Dothraki customs by leading her people without a Khal; why not break this tradition too? However, to shield her from Dothraki punishment (for breaking tradition), she simply chose to play dumb. She knew of the tradition, but had no intent to actually follow it. Rather than defy the tradition, she plays the role of the dumb foreigner. Especially since she is not Dothraki herself, and the Dothraki look down on non-Dothraki; Khal Moro is likely to believe that she is "stupid enough to not even know about this tradition!".
- Daenerys honestly thought this rule would not apply to her as she is not Dothraki.
- Daenerys thought the Dosh Khaleen was optional. Since Dothraki women accept (publically, maybe not in their minds) that they are considered lesser than men, Daenerys could have interpreted that the Dosh Khaleen are there for ex-Khaleesi who don't have anywhere else to go anymore.