The US government’s claims that Europe faces “civilisational erasure” because of mass migration are an attempt to score political points against Muslims, the home secretary has suggested.

Shabana Mahmood dismissed the idea that European civilisation and national identities were under threat because of migration and said the UK had managed the challenges of multiculturalism “very well”.

Her comments are the strongest response from a government minister so far to Donald Trump’s national security strategy. Published this month, it triggered alarm with its wide-ranging attack on European governments and suggestion that the US should promote “patriotic” political parties.

Asked by Trevor Phillips on Sky News to respond to the strategy and its implicit criticism of Muslims in Europe, Mahmood said: “Others will want to make political points about whether there’s too many Muslims in Europe or not. What I would say is we’re the sort of country that allows people to have the calling of their own conscience to live their own life free, but also has common rules that we all live by so that we live in peace together.”

The home secretary said she was “very proud to be a citizen of a country that is as diverse as we are” and added: “We are a multifaith, multi-ethnic country. I think that we have managed the challenges very well compared to other countries.

“That’s not to say that we don’t have work to do on integration, on the pace of change in our communities. It’s why I’m looking very closely at not just illegal migrants into our country but legal migration as well, to make sure it’s done in a way that retains public confidence.”

Trump’s security strategy calls for the restoration of “western identity” and claims that Europe will be “unrecognisable in 20 years or less” and faces the “stark prospect of civilisational erasure”.

It says the US wants “Europe to remain European” and “regain its civilisational self-confidence”. It celebrates the influence of “patriotic European parties” and says “America encourages its political allies in Europe to promote this revival of spirit”.

The strategy appears to endorse attempts to influence politics on the continent, saying that US policy should prioritise “resistance to Europe’s current trajectory within European nations”.

The UK government has been careful not to directly criticise Trump over the document and Keir Starmer, the prime minister, has said he would always stand up for Europe’s “longstanding values of freedom and democracy”.

Seema Malhotra, a Foreign Office minister, said in the Commons this week that although the government disagreed with aspects of the strategy, the US remained a trusted and valuable ally.

Sadiq Khan, who has repeatedly faced personal attacks from Trump, told LBC radio “we’ve seen in the UK a massive increase in anti-Muslim hatred”. The mayor of London added: “When President Trump says some of the things he does … he normalises and brings to the mainstream views that I think are unacceptable.”

Mahmood, who has spoken about how being a Muslim guides her in public service, said her faith was “the reason why I want to serve my country and do my best to make our country a better place for millions of people”.

Asked whether she thought the UK was ready for a Muslim prime minister, she said: “That’s not a thing that I can predict for the future but what I can tell you is you have a Muslim home secretary … and I think that shows the art of the possible in this country.”