I suspect the reason they've not used Mersey branding is simple: it's Liverpool City Region, not Merseyside. And LCR includes Halton which is in Cheshire. Metro is quite neutral and I wouldn't be surprised if they went with a neutral brand for political reasons. Now as I understand it, Halton's not in bus franchising at this stage, but as/when/if it joins in the future, they won't want to have to rebrand everything.
I think this is it in a nutshell.
In this respect, the neutrality is a benefit in that it can be readily applied and, as you say, it perhaps doesn't have the political issues of branding something Mersey-esque when some areas may not see themselves that way. As
@Djgr states, some on the Wirral might not see it that way, and neither in Newton le Willows or Southport.
The prefix Metro has been employed (to indicate a metropolitan or urban area) in many non-light rail instances; almost a synonym for "urban"
Blackpool Transport used Metro for their buses for a period, Bristol has the "metrobus" scheme, we have Metrobus in South London/Sussex though that might be a little incongruous when in East Grinstead

, minibuses in Cheltenham and Gloucester had the Metro brand in the 1980s/1990s, Metroline in London. There's doubtless others I've forgotten about.
Bus and train manufacturer Metro-Cammell branded various buses and coaches with Metrobus, Metroliner, Metrorider and the joint venture Metropolitan.