This argument is supported by various extreme-right and neo-fascist parties
Italy's national identity depends on the preservation of its traditions and life-style. If Italians allow immigrants to live among them, they’ll be contaminated by foreign cultures and will turn into a mass of confused people without a strong identity, easy to be dominated by those who hold the economic power.
Culture is a constantly-changing unicum, not a given item that must be passively took on. The authors of this argument also fail to remember that Italy doesn't have a fixed culture or identity - it's the result of many coexisting cultures and subcultures. Besides, it's not possible to have no identity. Humans are constantly shaped by the cultures and subcultures they inhabit. It’s good to preserve the memory of Italian traditions, but this doesn’t mean that those traditions must be continued, especially those in direct contrast with human rights (see the problem of traditional families against LGBT+ rights) or animal rights (Italy has a tradition of bullfighting, but nobody does that anymore). Also, how can anybody be sure that his/her grandparents' old life-style is better than others?
The co-existance of different cultures will destroy everyone's identity.
Rejecting the premises
The Italian language and culture already are the result of a co-existance of cultures, and identity is a constant process - it can't be "destroyed".