Is this sentence correct and why it's wrong?

The city today has become a center of education and a popular place for tourists.

The sentence is grammatically correct.

city / has become / center . . . and . . . place.

Punk is right, Wei, your examiner is wrong

1. 'has become' is Present Perfect, yet the sentence has 'today'. At some time in the past, educational institutions began establishing themselves here, so that over time (the period referred to by 'has become') it developed into a recognised "centre of education". Yet 'today has become' implies this 'becoming" happened TODAY, in contrast to the 'over a period of time' represented by the Present Perfect Tense. It should be:
'The city has become a centre… and a popular place/destination for tourists."
or
'By today, the city has become...
or
'The city today IS a centre…and a popular place/destination for tourists.'

I wonder about the use of 'place' in the sentence.
'place': a portion of space designated or available for or being used by someone
(It sounds like the city has been set aside for the use of tourists!)
Perhaps, 'a popular place for tourists to visit'…
…but in the given sentence, I would write:
'…a popular destination for tourists'.

All in all, I too doubt the competence of whomever set this exam: they have made a bigger error than their 'intended' error.

Today has another meaning besides "this day". It can also mean "this present time or age", particularly when making a comparison with a past or future time. This is the meaning used in the example sentence, where 'has been' is correct. The becoming happened in the present age.

Yes, today can also mean nowadays, and it frequently has that meaning.

The sentence is correct.

What has happened is that the bolded sample is correct and what's given as the erroneous example has been slightly changed.

I meant 'has become'.