So lets say you have are the owner of The Hotel Infinity with infinite rooms and in a single hour you defy physics and book an infinite amount of guests in the infinite amount of rooms. During the next hour, another person arrives and asks if there is room at the inn... of course the answer is yes. Because the size of the hotel in rooms is infinite, the subset of booked rooms is infinite... as is the size of available rooms booked rooms. There's always room at the Hotel Infinity.
Now, we can really have some fun worthy of the TARDIS: Let's say we have the pleasure cruise ship the S.S. Infinity, with infinite space for infinite guests and luggage. As your booking the infinite guests, you get a call from the Captain of the S.S. Infinity's sister ship, the S.S. Universe I which is built in the same class as the S.S. Infinity and to the same dimensions. He wants to book a stay on S.S. Infinity, and he's checking the entirety of S.S. Universe I (Including the passengers and their own cargo) as his own personal cargo. We can allow this because the S.S. Infinity has enough cabins for one more to be berthed and enough room in the hold to house an infinite amount of space, including a second ship of the same exact class.
Or better yet, a Time Lord can park his TARDIS inside another TARDIS, which itself is parked inside another TARDIS and so on and so forth.
If universe is infinite, it doesn't mean that a multiverse is impossible. It just means that the multiverse has infinite space for infinite subsets of infinite space. For example, the set of all real numbers is an infinite set... which includes the infinite set of all real numbers between 1 and 2. It doesn't matter to us how big the gap between those real numbers are (1.2-1.3, 1.245-1.247 1.2899999-1.9) there are are still an infinite set of real numbers between those gaps. Infinity in this sense doesn't care about value of a single real number, but the total amount of those real numbers within that gap.