Wiktionary gives the stroke order for 男 as:
I'm wondering if it's wrong to merge the two vertical strokes like this:
Question: Is it legitimate to write 男 merging the two vertical strokes?
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I think your question can only be answered if framed right, i.e.
When I write a given character this or that way,
and so on. One way to answer your question is the easy way out: just adhere to a well-known, published standard. In that case, there's clearly one right and many wrong ways to write a character, and @dan basically answered it: "The simple answer is 'wrong'. 男 is 田 + 力". Unfortunately, those prescriptive standards are often wrong or problematic themselves.
The more interesting way to answer your question is to look at how writers in the past and present actually do write a given character, and then decide whether you want to break or bend the apparent rules or rather follow them. In your specific case, I'd look at the many samples given at http://shufa.guoxuedashi.com/7537/1/ (kaishu), http://shufa.guoxuedashi.com/7537/2/ (xingshu) and http://shufa.guoxuedashi.com/7537/3/ (caoshu); I think what you can easily see there is that while there's plenty of evidence for a historical variant of 男 that has something like 𠠲 under the 田, there's not a single sample where 田 and 力 share the vertical stroke.
This leads me to answer your question with "you probably don't want to write 男 with a single vertical stroke because you'd be the only one to do so".
@nichijou's answer ("you break the consistency of writing any character with a 田") is generally reasonable, but then there are singular characters that are habitually or even prescriptively written in ways that are inconsistent, e.g. look at 所; it is often written like 𫠦 even in kaishu, and at any rate the left part retains the form 戶 although it should be 户 in the PRC.
Wrong, even if you want to be faster or lazily.
It is a convention that some strokes can be linked, some can not, some components can be reduced, some can not. If people break that convention in daily writings, he will not get a good impression from others.
If you don't know how, it is always a good habit to write strokes one by one.