Struggling but not stiff

Yes you can lift (or do any other exercise) daily if the intensity is of a level where you can still recover before the next training session. People often confuse recovery and rest but they are different.
If you aren't inducing more training stress (which it doesn't sound like you are) than you can cope with then yes you can train same muscles on consecutive days.

Actual numbers would help as I doubt you are doing a million reps.
What weights/equipment are you using?
Struggling to do how many reps when you go up an increment? (To a degree struggling/being challenging is part of the game,)

Maybe you could make your training more effective by changing what lifts/exercises you are doing or what equipment you are using (including bodyweight exercises)?

BTW - soreness isn't a training goal or even a good indicator of an effective workout. Getting stronger (higher weight and/or more reps is) over time is.

BTW2 - track tape measurements over time to judge muscle gain, if you are just starting out the majority of your increase in strength will simply be using your existing muscles better.

BTW3 - it does get easier as you get stronger as the increments become proportionately smaller. e.g. I'm rehabbing from an injury and when I was using 40kg a 5kg increase is 12.5%. Now I'm back up to 90kg the same increment is 5.5%.
Hang in there!

If I'm a female, would the tape measure measurement get larger or smaller?
Yes!
No idea if you are also losing weight - it's a way of tracking change over time.
(e.g. when I did a deliberate recomp adding muscle and losing fat at roughly weight maintenance arms and legs got bigger, waist got smaller, chest measurement stayed the same but that was a combo of bigger lats and loss of chest fat.)
Ditto progress pictures are helpful, with how gradual adding muscle is it can be hard to spot changes in the mirror (unless quite lean).

I did notice your comment that my first gains would be more in strength than size, which helps because I haven't seen much of a change.
Progress in weights lifted is the clearest indication your training is working and size is only one form of working.

What is the difference between rest and recovery?
Rest is a form of recovery but so is doing something different or reducing intensity / training stress.
(e.g. the day after a 100 mile cycle ride where I pushed myself very hard I did a gentle recovery ride to take the stiffness out of my legs and pump blood through well-worked muscles.)
As a long term high volume exerciser I rarely need rest (for me that need comes from complete fatigue not muscle fatigue) but I do vary my volume, intensity and exercise choices.
Someone new or returning to exercise may well need more rest but many also find active recovery more beneficial.

I don't tend to wait very long between different sets, which I would have thought means that I can increase my weight, but apparently not (based on me not being able to lift higher). It was mostly bench press.
Not resting between sets means your next set is being compromised, for strength training 2 - 4 minutes between sets is common. In your example where you are having to do loads of reps before making the next jump up in weights you could do short rest between sets to increase training stress (as noted by @nossmf ) but when you add weight, struggling to make the jump and your reps are low that would be a good time to fully recover between sets.
Yet another option is to alternate muscles being worked if you want or need to condense your exercise time (e.g. a push alternating with a pull, upper/lower....).

I wanted to throw my two cents in here to try to add some clarity from a different perspective of what's been said.

I would say that everything being said here is really great information so I'm not here to argue any of it but more to just piggyback on what's been said, particularly by @sijomial and @nossmf.

It looks as if you have a few different goals: 1) get stronger and 2) put on some lean muscle. Is that correct?

I think it's important to draw a line between the two of those and understand that they both play a part in helping one another but they are both different from one another as well.

There are millions of ways to do this so don't take what I'm saying as the only way. It's just for illustrative purpose.

1) Strength is more of a neurological function where your nervous system is learning to contract your muscles as hard as possible. Example: a 1 rep max deadlift. Think powerlifting.
Example workout:
Lifting 5 sets of 1 rep @ 90% of your 1rm resting 4 minutes between sets.
Example 4 week program:
1-5 sets of 1 rep @ 70% of your 1rm resting 4 minutes between sets.
2-5 sets of 1 rep @ 80% of your 1rm resting 4 minutes between sets.
3-5 sets of 1 rep @ 90% of your 1rm resting 4 minutes between sets.
4-5 sets of 1 rep @ 100% of your 1rm resting 4 minutes between sets.

There isn't much of a pump here and isn't really metabolically taxing either. The longer rest times are to allow the phosphagen system to replenish and the nervous system to recover so that you can repeat the effort again with success.

2) Muscle building is more of a metabolic function mixed with muscular breakdown. The main contributing factors to building muscle is: 1) TUT (Time Under Tension), 2) The eccentric component of the lift, 3) Progressive overload, and 4) Overall volume (to intensity ratio).
Example: Take bicep curls to complete muscular failure with a light to medium weight.
Example workout: 3 sets of maximum reps to failure resting 45 seconds between sets.
Example 4 week program:
1-1 set of maximum reps to failure resting 45 seconds between sets.
2-2 sets of maximum reps to failure resting 45 seconds between sets.
3-3 sets of maximum reps to failure resting 45 seconds between sets.
4-4 sets of maximum reps to failure resting 45 seconds between sets.

As noted by @nossmf there are lots of ways to switch things up to continue to add stress to the body to keep making progress.

I would also note that rest is something you do in the short term such as when you're resting between sets but still doesn't provide complete recovery where as recovery is something that's done over a few days or more where you're body is replenished with vital nutrients and your systems are made whole again.