This is not normal and should never happen. It is the result of improper technique. You need to find a teacher or, a better teacher.
There could be one of several things you are doing incorrectly. When you use two muscles at the same time to move one finger, or, when you press too hard into the keybed, or, if you play too much from the flexors, you can strain your tendons. Your tendons operate between your muscles and bones. When you flex a muscle it pulls the tendon which then pulls the bone, like a rubber band or pulley. The tendon is also encased in a sheath.
You could be creating inflammation (enlargement) of the tendon and it is stretching out the tendon sheath creating a tightness feeling. There is also a lubricant within the sheath called synovial fluid. You could be creating cracks in the sheath and the synovial fluid is leaking out creating an irritation. The lubricant could be forced out due to inflammation making it difficult for the tendon to glide within the sheath. The tendon can also adhere to the sheath due to lack of synovial fluid. When you move in the morning, the stiffness goes away as you "get the juices flowing."
If you are a weekend warrior, eh, you can probably play for thirty years but, TAKE HEED, this is the first, insidious and often ignored SYMPTOM of future injury. Other symptoms are cramping while executing trills, tremolos or octaves. It is because the pianist is using the wrong muscles and they ignore it because if shake their hands out, it goes away. But it didn't. It is CUMULATIVE.
It is like noticing uneven wear on your tires. You know they are wearing out but you think you can get another six months or year out of them. If you ignore them too long, you can skid off the road or one of the tires can blow. It wasn't just an accident, it was something which was becoming an increasing risk. You tempted fate.
Pianists do the same thing. They ignore these early signs of misuse until one day, something just breaks. They didn't suddenly injure themselves, they ignored warning signals for years.
Technically, you have early symptoms of tendonitis or, inflammation of the tendon. The next stage might be micro tears to the tendons resulting in the accumulation of scar tissue, which doesn't stretch. Tendons must glide and stretch. BTW, stretching can cause this, too. Many pianists are told to stretch. When you stretch, you are tearing tissue and warm blood rushes to the site of damage to begin repairs and it feels so good. But tendons lack vascularity and thus don't heal very quickly so the body places scar tissue there instead. Stretching = bad. I know, we've been taught the opposite for years.
Another symptom might be that the long flexor tendons become so inflamed that they press on the median nerve within the carpal tunnel resulting in Medan Nerve Entrapment. Also know as carpal tunnel syndrome but, it is MNE. Instead of fixing the problem of improper movement, most doctors and musicians treat the symptom of pain or numbness. We should fix problems, not symptoms. The problem is poor technique.
I'd have to see you play but the most common error pianists make is either pressing into the keybed, playing from the flexors instead of the elbow and gravity or, the MOST common error: abducting the fingers. You can't flex and abduct at the same time. Well, you can but shouldn't. When you use two muscles to move one bone the tendons play tug of war with the bone and one of them or both will strain. We feel cramps or tightness and just shake it off. Another symptom is uneven playing. Often, poor playing doesn't mean we lack talent or need more practice, quite often we are moving incorrectly and getting in our own way.