Considering that there isn't a specified amount of time between the end of one year and the start of the other. That is to say, the end of one second and the start of another second happen simultaneously, an indistinguishable change in which only the ends, and the middle, are observable. So for all practical purposes, he is technically correct.
Similarly the end of one year is decided by one full revolution of the Earth around the Sun. Such that the revolution for one year ends seamlessly into the next rotation, just as it always has.
Nobody can answer whether 1895 ends at the exact same point 1896 starts, because nobody knows whether time is ultimately discrete or continuous (I mean fundamentally, can you get ever decreasing increments of time, or is there a point where time becomes digitized in nature?).
If time is ultimately discrete, then 1895 ended before 1896.
If time is ultimately continuous, then 1895 ended exactly as 1896 started.
I think that it is thought that there is no singularly small unit of time, we just derive it from the rate at which certain things happen.
However, if time is derived from an orbital, geometric shape, then we can have a good estimate. Distance may seem similar to time, but we can say that the smallestt unit of distance is something like a subatomic particle. We might call the rate at which the planet travels that distance 'the smallest unit of time possible for measuring the speed of the planet' (I'm not good at snappy names). On one side of that is 1895, and on the other is 1896. so at true 00:00000000 etc (or the closest that we can derive) which is so tiny as to be negligible for every purpose we may say that 1895 ended at the same time as 1896 began. The realist answer would be that time is infinitely linear (how long is a piece of string), the functionalist answer would be mine and the idealist would say that time is an illusion, it doesn't exist.
Perhaps a better way to say it is that 1896 ended 1895. Neither can really be in one another, the moment that 1896 began is the moment that 1895 ceased to be. No overlap.