Nope. Unlike in Russian, both can be used. One expresses possession through the "existence" of something, the other just uses the verb «мати»(to have).
Some say, «у мене є» is "worse" because it is a calque from Russian «у меня есть», and should not be used. Which is, AFAIK, not supported by the reality of Ukranian literature in the last two centuries.
The current use is:
- in Ukranian, both "Я маю" and "У мене є" are used. I cannot comment on their relative popularity, but, it seems, both are popular.
- in Russian, "У меня есть" is the only widely used form, "Я имею" having its use restricted to set expressions and official style (it also has a sexual connotation "I screw").
Naturally, if the influence of Russian increases (the languages are pretty similar, grammar-wise), a Ukranian speaker will tend to use "Я маю" less, even though in Ukrainian there is no stylistic awkwardness associated with the verb "мати".