Yes, it is 100% improper, even in Scotland and other countries where clans exist or have historically existed (which doesn't include either Finland or the USA). Your friend is a fool.
A surname is, essentially, an extra identifier to distinguish a person from all the other people who might have the same personal name, and to indicate their kinship with their family. However, many surnames have been adopted many times over by totally unrelated families: for example, just about every village in medieval England had a smith, who was an important figure in the local economy and whose family were therefore highly likely to be identified by the surname Smith. This is why the surname is now so common. So having a given surname, let alone adopting a given surname, does not automatically create any relationship with everyone else who happens to have the same surname.
Clans are something else altogether. In medieval Scotland, where law and order were frequently nonexistent, people joined together in groups under a chief, promising to obey the chief in return for protection from anybody else who might hurt or rob them. The clan was imagined as an outsize family with the chief as the family head, but in fact you didn't have to have a blood relationship with the chief to be a member of his clan; you just had to promise to obey him and fight for the clan when necessary. Nor did you have to take the clan name as your surname; many Scottish clans have always included people with a range of different surnames.