The OP's second sentence is grammatically correct in standard British English
It is twenty years since Arsene Wenger became manager of Arsenal.
Some examples taken from the web
How has Russia changed under Putin?
- It is more than 15 years since Putin became Russia’s president. The former KGB officer was a virtual unknown when he was picked to be prime minister by Russia’s first president, Boris Yeltsin.
23 July 2015, The Guardian
From another British paper, the Independent
- It is 10 years since UN peacekeepers were killed in southern Lebanon – and it could happen again now
From the BBC website
- It is 200 years since the birth of Charlotte Brontë, who in Jane Eyre created one of fiction’s most memorable heroines.
From an Irish newspaper, Independent.ie
- It is 20 years since scientists announced the world's first successfully cloned sheep, Dolly.
And in a Wikipedia page dedicated to Scientific opinion on climate change, it quotes the UK Royal Society
- The society says that it is three years since the last such document was published and that, after an extensive process of debate and review, the new document was printed in September 2010.
The main verb in the OP's sentence is become, the act of becoming something or someone may require hours, days, weeks, or, in extreme cases, even years. But in the original sentence, it didn't take Arsene Wenger years to became manager (coach) of the London football team, Arsenal. He was offered the position and he immediately accepted it. Thus the Simple Past tense is used, and was used in all the above examples. The original statement can be rewritten as:
Arsene Wenger became manager of Arsenal twenty years ago
If the OP wants to convey that Wenger still holds the position, then the main verb has to change, and the Present Perfect tense should be used.
- Arsene Wenger has been manager of Arsenal for twenty years.
- Arsene Wenger has been manager of Arsenal since 1997
- Twenty years have passed by since Wenger became manager of Arsenal.