Well, I can't speak to Chinese as I don't speak it even a little.
But if that's a claim that Chinese has no tenses, I guess it somewhat comes down to what you consider a tense, then. Because if it's just about forms of a verb, I'd have to say that English has no future tense, for example, leaving us with probably... three tenses? Present (do), past (did), progressive (doing).
And "what you consider a tense" is semantics. Full circle.
To me a tense is the way you normally and automatically construct the grammar of a sentence to implicitly convey a specific idea about time. Without saying it explicitly. So "I went to the store", "I went to the store yesterday", and "I went to the store last week" are all the same tense. Because you've communicated "past" through the grammar without having explicitly said it. And it is clearly different than "I am going to the store" or "I will go to the store".
The Chinese form, by contrast, all look like, "I go to store". And the words saying whether it was somewhere in the past, yesterday, last week, now, or some time in the future need be nowhere in the sentence. It is enough that they were explicitly said at some point.
And this is specifically something that Chinese people struggle with in English. You constantly will catch them making mistakes like, "I go to store yesterday." Because the idea of conveying time with grammar is as foreign to them as the idea that objects have gender is to English speakers.
But if that's a claim that Chinese has no tenses, I guess it somewhat comes down to what you consider a tense, then. Because if it's just about forms of a verb, I'd have to say that English has no future tense, for example, leaving us with probably... three tenses? Present (do), past (did), progressive (doing).
But nothing I've read treats tense that way.