As written, the sentence assumes that only two brands of toothpaste exist. This sentence calls for the superlative form "most."
"more" vs "most" does not come into it. "prefer" is enough, and "more" can be cut.
Agreed. You can like something more or most, but prefer stands alone. It works equally well whether you a choosing among 2 or 200.
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prefer indicates that a person is MORE INCLINED to take a particular brand of toothpaste than another brand. "More" is redundant and superfluous.
"I prefer this brand of toothpaste" taken alone may make sense in context - but really even that doesn't express a whole thought. Consider that out of context you have to ask:
"Prefer it to what?"
Then you might get the answer:
"I prefer it all the other brands" or:
"I prefer it to being beaten with a hammer."
Again, consider: "I prefer coca-cola." Presumably, to other soft drinks (as this is the normal context) but maybe other brands of cola. Out of context, there is not truth-evaluable statement here.
So - now "more". Again, any sentence with "more" in should contrast two things. If you say "I like this toothpaste more" again we need ask; "more than what?" and the same options apply.
At any rate, to have them both in here, combined in this way, and out of context, is surely poor English.
It's clear saying as,
I prefer this brand of toothpaste more then other brand.
What's wrong with it?