Hmm. You *could* but the connotation (underlying meaning) is a little different depending on the sentence structure.
'My partner is called Jane' - This is a fact - her name is Jane.
'I call my partner Jane' - This may not be her real name, but it is what I *always* call her. Perhaps her name is January [which would be an awful name]. But it implies that she accepts the name-change.
I call Jane my partner -- I refer to Jane as my partner
In this case, both of these sentences would be the same. However, this sentence (specifically about partners) would be very odd sounding to a native speaker. It almost implies that *I* call her my partner, but *SHE* doesn't call me her partner.
Often when 'call something X' and 'refer to something as X' are equivalent is when we're implying that it isn't the 'true' name, so we'd usually say in a casual conversation...
I *like* to call her my partner. / I *like* to refer to her as my partner.
In this case, the two would be identical.