Because of comments to my first answer and the fact that I personally think I only suggested options that require dices being rolled, which inherently includes a risk of the player (or panther) actually failing and dying, I'm gonna give you some more suggestions as to ways of 'shining the spotlight' on the Animal Companion that do not imply risking its life (no matter the fact that it can always be brought back).
Remind the ranger and the rest of the party of the Panther's existence
This can be done multiple ways.
There are passive ways of doing it. Mention that your character falls asleep petting the panther around the campfire, when you get the opportunity.
Or mention that you hate when the panther brings dead animals to your bedroll, when you wake up one of those mornings ... if you want to develop a more conflictual relationship with the panther (and its ranger).
A more active way of doing it is by including PCs other than yours or the ranger, by reminding them to buy food for the panther when they are sent to go shopping for supplies. Or ask the ranger what is the panther's favorite snack, if you go hunting for food.
Create RP opportunities by simply "reminding" the group that there is a panther in their midst, basically. Chose your moment and do this for all characters in your group and soon you will find that everybody joins in and thus a real group dynamic is created.
The ranger should use the animal companion as support
Maybe the ranger class really is the under powered class everybody says it is. But I still think that, with the animal companion's ability to use the Help action on command, the ranger really has many opportunities to shine, especially out of combat.
Having the panther constantly assisting the ranger can make the ranger VERY efficient at all her skill checks and attacks.
Even though it is not a given, a DM could rule that having the ranger use 1 action to ask its animal companion to help her follow tracks/sneak/get through those vines is not a command that needs to be repeated every round ...
... which can make the ranger very good at 'skills'. With stealth, this makes for a very efficient scout. With survival, this makes for a very efficient tracker and huntress. This can be applied to almost all skill checks a ranger would want to make, except knowledge checks. Potentially, depending on the DM.
Also, do not forget the fact the animal companion is expandable. It can be a life saver that rogues never get when they go scouting. It can be sacrificed to gain precious seconds and allow a ranger to survive where a lot of others would die, especially at lower level. When worst comes to worst, this is a very versatile way to deal with the situation.
Get the ranger to actually play to her nature/strengths
A ranger is supposed to be wild. The fact that it actually befriends a panther is proof of that.
Would the ranger and her panther let themselves be replaced by a bard and a flying kitten ? This might be an opportunity for a 'friendly' competition between pet tamers.
Or maybe the ranger and the panther actually fell in love with that flying cat ? The panther and ranger might end up as the perfect protectors for that other feline ?
Or maybe the panther decides to just try and eat the kitten, thus forcing the party to get into all kinds of shenanigans to try and recapture it ? A DM could also have a lot of fun with this kind of situation, created by the ranger's personality/reaction to the world surrounding her.
Create your own inside jokes centered around the panther
This is more of a OOC way to create shared history in a group, but it is just as effective as other ways, when done right.
Maybe the ranger will succeed at everything it tries to do by being lucky on dice ? Make jokes about the panther being a lucky charm or something.
Maybe the ranger will fail everything due to bad luck ? Just keep muttering about how "black cats bring bad luck" until the ranger's player actually decides to change her animal companion by getting supersticious !
In the end, most of the responsability of making the ranger + animal companion combo effective and/or fun is actually up to the player, the ranger.
Other players can help, by interacting with the animal companion in character or talking about its successes or fails... but in the end, it is also a lot about how the ranger itself accomplishes by itself what we can only hope will lead to good fun.