Eta was appointed interim head coach on Saturday for the remainder of the season following the dismissal of Steffen Baumgart with Union in 11th place.
She will oversee the men’s first team through the end of the campaign before moving to the club’s women’s side.
"I totally understand that in public there is this issue (of the first woman as a Bundesliga coach)," she told a press conference ahead of her first game on Saturday at home to strugglers VfL Wolfsburg.
"I obviously know the issue and it was never about that. It's always about football, about working with people ... and about being successful.
"I have the trust of the club and I am happy about it. The most important thing now is to prepare for the weekend against Wolfsburg," Eta said.
Asked if she had any hesitation when getting the job offer last week, Eta added: "There was nothing to consider. I was at home on Saturday so I was surprised initially (to get the call) but then I was happy for this trust, and then things happened quickly."
Eta said she was by no means an exception in football with women taking over more and more key positions in many clubs around the world.
"There are female coaches out there before me. In many other countries as well," Eta said. "I understand the issue and the interest in it but it would be nice if this issue did not exist and it was only about success on the pitch."
Eta has been coaching Union's Under-19 team.
She became the first female assistant coach in the German top flight with Union in 2023 but her appointment this time has also triggered some derogatory social media comments, which have been condemned by the club.
"This (social media comments) does not concern me because it says more about those posting these things than about me or other people they are talking about," she said.
"That my appointment became so big may have surprised us a bit because for us it was not really. I hope in the next few years this will be less and less and people will focus on football."