I hold both HIVE Stock in my TFSA (tax free savings account in Canada) on Wealthsimple and HIVE crypto on exchanges as well.
Hive stock is listed on open markets and they probably have to make an attempt to sue any breach of brand recognition etc even if they don't follow through it may discourage other online or brick and mortar companies in related industries to run with the same name. I would argue that it is probably wise to do especially for a company with listed stocks as the board have shareholders to answer to.
Since 2017 it was listed on the Canadian exchange as HIVE.
With regards to changing the ticker in America other than consolidating the ticker to match Canada, there have been studies that a portfolio of readable/memorable tickers outperform a bunch of letters. i.e. using HIVE as a ticket is better than HVBT everything else being equal.
I'm not sure if HIVE crypto had any positive or otherwise impact on HIVE stock. I don't think a small cap crypto is changing the price on a larger cap listed stock in any meaningful way.