Are bears related to dogs in a similar way to how cats are related to 'panthers'?
No, because panthers are cats, but bears are not dogs. You can see their phylogeny in a figure from one of my papers here: https://www.researchgate.net/figure/Most-likely-phylogram-derived-from-the-Bayesian-Analysis-ln-533753675-Numerals_fig1_6550244

As you can see, Bears are most closely related to pinneped (seals and sea lions), with dogs as sister to that group.