Put on hold and put on weight should not be analysed similarly.
There is more cohesion between on and hold in the first example than between put and on, and the expressions put on hold and on hold with the same sense are both given at McGraw-Hill Dictionary of American Idioms and Phrasal Verbs. © 2002 by The McGraw-Hill Companies, Inc. There is no reason to consider put on as a multi-word verb here. On hold is an idiomatic prepositional phrase, and put [somebody / something] on hold is also an idiom.
However, on weight is hardly unitary, and is not found in the above reference work. Put on here is however correctly analysed as being unitary - a MWV with the meaning 'increase (in)', 'add', 'gain'. Put on has several senses as a MWV, listed at thefreedictionary (where they use the traditional term 'phrasal verb'). I wouldn't regard on as a preposition here (though some would) - I prefer just to see MWVs as verbs (even though they might sometimes surround their objects!)