Two dummy variables are mutually exclusive

Hello all,

I have a problem for a university assignment that two dummy variables are mutually exclusive.
The task is to estimate what can increase certain diseases in dogs using regression.
In the sample there are several factors (Lives in city yes/no, age, male/female etc....).
Then there is a variable owner, divided into male, female, couple.
For each of these, a dummy variable is to be created.

If I run the regression with couple and man, I get a coefficient of 0.0545 for man.
However, if I run the regression with couple, man and woman, I get for woman - 0.0545 and the coefficient for man is 0 and for the standard error I get "omitted".
So I get the exact opposite effect for man and woman respectively.
What do you call such an error and what can be the reason for it?
What do I also have to consider later in the interpretation?

Many thanks in advance!!