In the Ottoman Empire, Christians were allowed to practice their religion freely, though I'm aware there were plenty of other restrictions and limitations imposed on them. Were there any contemporary Christian kingdoms with a similar policy toward Muslims?
Also, and perhaps more generally, were there any Muslim populations at all in Europe in the 600 or so years that the Ottoman Empire existed?
I'm asking about the Ottoman period because I made the horrible mistake last night of trying to watch a film called Day of the Siege on Netflix, a godawful production that is best avoided. The film purports to tell the story of the Battle of Vienna, between the Ottomans and an alliance of Central European states. I turned it off early (it was that awful) but it prompted me to think up these questions.
On your second question, there were certainly Muslim populations in Europe during the Ottoman period. The Ottomans controlled the Balkans, Greece and much of Eastern Europe. While not all populations became Muslim, Bosniaks, Albanians and Kosovars remain majority Muslim to this day. After the collapse of the Ottoman Empire, the Muslim populations of Greece were forcibly removed in 1923 by treaty during the population exchange between Greece and Turkey.
You may be interested in reading up on the devsirme system, which converted Christian boys to Islam to build up the Janissaries corps.
On the other side of Europe, Muslims were in Spain until the conclusion of the Reconquista in 1492. So there was a brief temporal overlap with the Ottomans.
Even the adversaries of the Ottoman had Muslim populations the one that springs to mind to me were the Lipka Tatar who were living in the Polish-Lithuanian Commonwealth from the 14th century until today. They served Poland as highly effective light cavalry through the Second World War, at their peak in the 1590's there were an estimated 200,000 in 400 mosques. Over time they became increasingly Polonized adopting Polish or Belarusian language and culture (but retaining Islam as their religion). The World Wars and Communist period was very hard on them and the according to recent census 10,000-15,000 remain in the territory of the former Commonwealth.
About the Siege of Vienna in particular, Lipka Tatars fought in the army of Poland in the breaking of the siege. They wore a sprig of straw in their helmets to distinguish themselves from Tatars fighting on the side of the Turks. It is traditional for Lipka Tatars to wear straw when visiting Vienna to this day to commemorate the heroism of their ancestors there and in saving King Jan III Sobieski on the first day of the Battle of Párkány.
Following the Swedish king Charles XII's stay in Turkey after the Swedish army lost the battle of Poltava and as his attempts to gain a Turkish alliance in his wars against Russia failed he returned to Sweden in 1714 along with a small group of muslim merchants and soldiers.
The king had become heavily indebted while staying in Turkey and these merchants were debtors looking to collect from the Swedish state.
They are notable because at that time it was illegal for practicing muslims to live in Sweden. Sweden was strictly and exclusively Lutheran and in that sense muslims were no more welcome than jews or catholics. Nevertheless, this small group of Turks became a tiny muslim community in Sweden. From what I read most merchants eventually settled their claims with Sweden and returned to Turkey, though not easily because Sweden was ruined by Charles' wars, and the remaining Turks converted, but for a while Sweden did have a Turkish presence in Stockholm.
The law never officially changed during their stay, Lutheranism was the only allowed religion until 1741 (when they graciously made allowances for Anglicans and Reformed churches), but politics and state debts can make exceptions.