I don't know where you got that "lie" from. Posteo doesn't allow custom domains in order to have as less information about the customer as possible (what it calls "data economy", meaning it doesn't want to store data wherever it can be avoided). I have seen the FAQ multiple times, where Posteo has had this answer for not allowing the use of one's own domain. [1] I don't like that Posteo doesn't offer custom domains, even if someone is willing to make the tradeoff that Posteo believes it to be. But I don't see anything wrong in its ideological position.
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> Can I use Posteo with my own domains?
> No. We are an email provider with a particular, privacy-oriented model – and this is not compatible with incorporating own domains. One of our emphases is data economy: we do not collect any user information (names, addresses, etc) of our customers. We always answer requests from authorities for user information in the negative. On the other hand, own domains need to be registered to the name and address of a person. If you were able to use own domains with us, this would affect the entire concept of Posteo: we would need to start saving user information for all customers who use their own domains with us – and to provide these to the Federal Network Agency to be provided on request to the authorities.
> Even if only the MX record pointed to us, we would still need to store the assignment of the domain in your Posteo account as user information. Thus we would possess your user information and be required to give it out. For this reason, we have decided not to offer this possibility and instead to use data economy. We certainly understand that having your own domain is very important in the commercial industries, but from our privacy-oriented perspective, the disadvantages prevail. It is, however, possible to add various other email addresses with external domains as senders in the webmail interface and thereby to send emails with Posteo using external domains. In order to be able to read replies to these messages, you need to set up forwarding to Posteo for the external address.
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Also please see its stance on privacy [2], where it emphasizes minimizing the data collected from the customer.
[1]: https://posteo.de/en/site/faq
[2]: https://posteo.de/en/site/privacy