Does any other country besides the USA have states?

It depends how you define "state".

1. If we are going to say that any country divided up into administrative regions counts regardless of what they are called then the answer is that MOST countries do.

2. If we are talking about regions with a federative connection:

Argentina, Australia, Austria, Belgium, Bosnia and Herzegovina, Brazil, Canada, Comoros, Ethiopia, Germany, India, Iraq, Malaysia, Mexico, Micronesia, Nepal, Nigeria, Pakistan, Russia, Saint Kitts, Somalia, South Sudan, Sudan, Switzerland, UAE, USA, Venezuela

https://en.m.wikipedia.org/wiki/Federated_state

* Note that I am drawing a dividing line between those places that have a federative connection (as in USA, Germany, UAE, Malaysia, Mexico, etc) and those that don't (the UK isn't a federation of counties, nor is France a federation of departements). If it has a government, it's a state. If it just has a council, it's a county (or equivalent).

Federated States of Micronesia, which is in Oceania.

I don't think counties count. States have a certain amount of independence and lawmaking ability which counties don't. The Swiss cantons are amongst the most independent in theory, but the states making up the United Arab Emirates and Malaysia have a considerable degree of independent action as well. Law-making (as opposed to byelaw making) is found in the German Lander. The UAE and Malaysia situation is somewhat akin to the UK in that most of the states are monarchies, but unlike us in that they have different monarchs (not one for the lot). (The Yang di-Pertuan Agong is the monarch of Malaysia as a whole, chosen on a rotating basis from the sultans of the individual states for a fixed term.) Catalunya in Spain could be regarded as a state now that it has a degree of internal autonomy.

The key feature of a state (whether called "state", "land", "province" or "canton" or whatever, isn't that it's an "administative" entity. It has a considerable degree of autonmy that is protected by law: England has counties but they can be merged or sub-divided at will by Parliament without further ado. Swiss cantons and the German laender - and surely also American states - can't be treated like this by the national legislature.

What may be the features of a state in the USA may not be the features of a state in Mexico, India or Australia. The word state is an imprecise term with a dozen or so meanings which may or may not imply sovereignty, autonomy or whatever. I think that each country has its own historical reasons for calling its districts states, provinces, Lander, cantons, counties, republics or regions. Because of this it is almost impossible to find one strict definition for each of these words that only applies specifically to the places so described.