Can I say :” make something five meters higher “ or “ make something higher by five meters “ ? I mean Can I use “ five meters “ in these ways ?
Yes, both of those uses are grammatical. The first is a bit more natural to a native speaker, but both make sense.
In some cases, though, you might want to use "Five metres taller instead. You'd normally use "Five metres higher" if you were speaking of something abstract, like a distance to be climbed or jumped, or "Five metres taller" when speaking of something discrete and physical.
"We need to make the rope climb five metres higher."
"We need to make the rope climb higher by five metres."
"We need to build the wall five metres taller."
You might also use put instead of make if you're speaking of placing a single item at a specified height:
"We need to put the basketball hoop five metres higher."
It's certainly possible to say, "Make it higher," for objects placed at a certain height -- but it sounds rather juvenile, as if the person didn't know the verb "raise" (or its synonyms).
Raise it by 5 meters please?
For objects of a certain height, which you want to increase, the preferred expression is, "Make it taller".