Can a New Game + player summon or be summoned by a New Game player?

The only thing that affects summoning in Dark Souls is your SL (Level) range. Your NG+ status doesn't affect anything in terms of multiplayer. There is no 'official' list that I know of for SL ranges for multiplayer, but there is a calculator you can use online(for both Remastered and original Dark Souls) to find out.

Please note that there may be a +-1 level difference in reality from the calculator because it's fan made.

Soul Level 123:

You can summon levels 101-145.

You can be summoned by levels 103-148.

However, Dark Souls Remastered added a new aspect to the multiplayer formula: Weapon Matchmaking. I suspect this is where your problem lies.

Now you must have a similar weapon level to the other player you are trying to play with, or you won't be able to join up. Refer to the weapon level calculator by scrolling down here to see if that's what giving you trouble. At the maximum weapon level of 15 though

Weapon Level: 15

You can match with weapon levels 9-15.

So if you've got a maxed out weapon you may need to switch it up if your buddy is using something lower-level.