The "tmobile" phone sold in the US Apple store is unlocked. They merely provide a tmobile SIM for "convinience". You can throw the tmobile SIM away and use your own, or activate the tmobile SIM if you wish.
Apart from having no warranty in the EU (or anywhere else, except the US), LTE service might not work and Apple has a disclaimer that it is not guaranteed to work outside of the country the phone was sold in.
About the unlocked iphone 5s:
If you don’t want a multiyear service contract, or if you prefer to use a local carrier when traveling abroad, the unlocked iPhone is the best choice. If you choose T-Mobile, your iPhone comes with a T-Mobile SIM card already installed. You will need to contact T-Mobile or visit an Apple Retail Store to activate your iPhone.