How can I increase the power capacity of a kitchen circuit?
Know your appliances
Every one of your appliances has a nameplate or something stating its power usage.  You are looking for any of the following:

Amps (typically 5-13)
Watts (typically 200-1500)
VA (typically 200-1500)

If you get watts or VA, divide by 120. That gives us amps.
Expect most of your kitchen heat-making appliances to be 1500W, or 12.5 amps.  This is because of the UL White Book, the safety standards for making appliances: UL only allows 1500 watts from appliances which plug in.  Sometimes a microwave will be permitted 1800W, since it is guaranteed not to be run continuously.   1800W is 15 amps.
Know your power source
Look closely at your power strip.  It will state a maximum amps that it is rated for (or again, watts; you know how to convert).   I would expect 900-1800 watts (7.5 to 15 amps).
Now, go to your circuit breaker panel. Look for the circuits which feed the kitchen.  There will be a number of funny looking switches, most of which say &quot;15&quot; or &quot;20&quot; on the handle.  See if you can identify the fridge circuit and the kitchen circuit you normally use.  The number on the handle is amps - that same figure we've been working with.   That is the absolute hard-maximum capacity of the circuit for intermittent use.   If you're running continuously (3 hours or more) you should only use 80% of that.
Balance your appliances.
The nameplate is accurate on hand-held plug-in tools.  It's not so accurate on certain appliances and I'll cover them.

Refrigerators take about 1 amp when running.  A bunch more for an instant while starting, but 1 amp once running.  That's not much at all, is it, compared to other appliances?
The gas oven takes even less - 0.1 to 0.5 amp depending on if the oven light is on.
A panini maker runs at the UL limit of 12.5 amps. PIGGY!
A small electric oven runs at the UL limit of 12.5 amps.  PIGGY!
A microwave runs 12.5 to 15 amps, depending on what UL has allowed.  PIGGY!

Now, if you're doing the math, you just noticed that the oven and refrigerator Do Not Matter.  However, the three &quot;piggies&quot; each take 12.5A or more.  There's no way to run 2 of them without pulling 25 amps, and  that's too big for everything - too big for the power strip, too big for a 15A breaker and too big for a 20A breaker.
It ain't gonna happen. You cannot run 2 appliances at once on one circuit.  A different power strip cannot fix this. The circuit breaker downstairs will just trip instead.
What about that refrigerator circuit?
Modern Code requires that every kitchen have two 20A receptacle circuits.  I don't know if your kitchen complies.  If it does, the other circuit might be that refrigerator circuit.  Find Out.  Turn off the circuit breaker that feeds the oven and range, and see if the fridge loses power too.
If the fridge is on a separate circuit, well, fridges take very little power.  It can easily share the circuit with ONE - exactly one - piggy.
The reason we prefer to dedicate fridge circuits is we don't want another appliance causing the circuit to overload or ground-fault, and trip out the fridge while you are away or without you knowing it.
Also, it is vital that small kitchen appliances have protection called GFCI.  That protects you if an appliance falls in the sink, or has an electrical problem of some kind. That protection is useless for refrigerators - you're not likely to drop one in the sink! And it can spoil food if it has a false trip, which happens a lot with refrigerators. So fridges are often not on GFCI.  So if they told you not to use that circuit, see if that is why.  If the plug is more than 6 feet from a sink, it should be alright to use.