When to use counters with 这 and 那
You don't necessarily need a counter or classifier in that case.
這個是什麼 is ok, it's like "What is this thing?"
這是什麼 is also ok, probably more common. "What is this?"

那是我的自行車 is perfectly fine, just a generic "that's my bicycle"
那個是我的自行車 is like "that one is my bicycle." (Not this one or those other ones)
The counter word just emphasizes that you're talking about that particular thing.

It's not mandatory here because there is already 我的 that classifies the bicycle.

It's also unnecessary if you're talking about a general class of object, rather than a particular object. 這是什麼？這是自行車. Or 我要買自行車 "I want to buy a bicycle" (not any specific one, just one in general)

If you said something like 給我那個自行車, then the classifier would be mandatory, because you're talking about a particular one, so you need to specify it. But 給我你的自行車 has 你的, so it doesn't need a counter word.