Is it legal for my employer to make me pay employee NI and employer NI?

Absolutely not legal, unfortunately a lot of these so called umbrella companies are total cowboys. If you are an employee you pay employee's NI. If they are cling you as self employed some of them withhold sufficient funds to ensure that you pay your liabilities at the end of the year. One thing they all have in common is that they charge extortionate rates for the mediocre service that they offer.